What you need is: Two plastic disks, with about 31 cm of diameter (It is possible to use exactly two old LP disks, but the result is rather ugly). I used two acrylic disks, about 2 mm thick, with 20 cm of diameter This is rather small, but big enough to allow the observation of all the important electrostatic phenomena, without taking too much space. If you want some power, make disks with more than 30 cm. The spark length that can be obtained is about 1/3 of the diameter of the disks.
Mount in the middle of the disks wood cylinders, to be used as "bosses" to turn the disks, with precise holes to pass a steel axle in the center. Make grooves around the cylinders to be used as pulleys where the cords that will move the disks will pass. I actually used more elaborated bosses, turned in a lathe, with flat faces to be glued to the disks in one side and small pulleys at the other side. Metal or a plastic as nylon can also be used for the bosses. Mount the assembly in a wood support composed of a base and two upright supports, in a way that allows the disks to rotate in opposite directions, maintaining a separation of 1-2 mm, never touching. The distance can be proportionally larger with larger disks. The construction must be solid and well balanced. I fixed the bosses to the disks with glue, adjusting the disk positions while the glue was drying. Larger machines require screws to fix the disks to the bosses. I like to use three flat-head screws with a flexible washer between the disks and the bosses, so the pressure of the screws can be adjusted to make the disks run true. Ball bearings between the bosses and the axle are a good idea. I used some cardboard washers soaked with paraffin to adjust the spacing between the disks. It's important to select the plates for the disks with very uniform thickness, or they will vibrate when turning fast. If this happens, it's possible to glue small lead blocks to the edges of the disks, at the lighter sides. Not a perfect solution, but works.
Mount in the same support two pulleys, larger than the boss pulleys, in an axle moved by a crank. This axle can pass a few cm below the disks, mounted on adequate bearings. For this small machine, I just made holes in the upright supports and inserted a layer of brass foil between the axle and the wood. In other machines I used brass, bronze, or Nylon for the bearings. Ball bearings are the best solution. Pass cords from these pulleys to the boss pulleys. Turning the crank shall make the disks turn in opposite directions at several turns per second. Cross one of the cords to make one of the disks turn in the opposite direction. I used rubber cords of the type used in tape recorders to connect the pulleys to the disks. Large "O" rings make good cords too, but that don't last much. Round leather cords are the classical material, but sewing machine cords are too thick for a machine of this size. An exellent material are polyurethane cords that can be joined by melting. When joining the cord that will be crossed, make two loops on it. In this way the cord ends with a half twist, and operates better. For the other cord, make just one loop.
Make a set of aluminum sectors from thin aluminum foil. The sectors shall be perfectly flat stripes a few centimeters long, with one side larger that the other, and with rounded corners. A minimum width of about 1 cm is adequate. These sectors are to be glued to the plastic disks, at the external face. forming a symmetrical pattern around each disk. The number of sectors shall be even, so there are always two exactly opposite. More sectors is better than few, with the usual number being between 16 and 40 (I used 18). The output current of the machine is proportional to the area covered by the sectors. The maximum spark length that the machine can generate can be estimated as the sum of sector spacings along a third (for the neutralizers at 60 degrees) of a disk. A distance between sectors similar to their average width is adequate (they have the outer side larger than the inner side to keep this fixed distance along their length). Wider sectors result in more current, but in smaller sparks. The distance from the sectors to the disk bosses also limits the maximum voltage. Originally, I used kitchen aluminum foil for the sectors. To glue the sectors to the disks, I used common paper glue, soluble in alcohol, fixing tightly the strips to the disks, with help of a piece of cardboard, leaving no bubbles. The excess of glue I removed carefully after it dried with alcohol, leaving the disks perfectly clean. It is important to do not leave any sharp corner in these sectors. The thicker foil used in discardable food containers (a pizza pan is ideal) is a good material for sectors, and is more resistant to wear. In this case a stronger glue is required. A "contact" glue based in synthetic rubber is adequate, and results in a clean disk that can be used almost immediately. A very convenient possibility is to make sectors using adhesive aluminum tape, of the type sold as "metal repair tape". Look for a type that has a backing foil, that simplifies the operation of marking and cutting out the sectors. To make the sectors, make first a hard cardboard template, a bit smaller than the desired sectors, use it to mark the metal, running a pencil around it (hence the smaller size). Cut the sectors carefully with scissors. To apply the sectors with precision, it's enough to make pencil marks on the disks, or to draw a template that is kept under transparent disks.
Adapt to the upright supports two solid wires having at the extremes very thin flexible metallic brushes (I used just one thin wire as brush) that touch the disk sectors at opposite sides of each disk, at adjustable 45-60 degrees angles with the horizontal and at crossed positions. These "neutralizer bars" shall short-circuit two opposite sectors when they pass under their brushes. Thin silver foil strips is the ideal material for the brushes. The thin nickel-chrome wire from a high-value wire-wound resistor can also be used, thin enough to not scratch the disks or the sectors. The brushes can be fixed to the neutralizer bars with a section of plastic wire insulation, can be inserted in holes at the ends of the bars, or some other form that allows simple replacement of broken brush wires. The neutralizer bars are fixed to metallic rings, fixed to the upright supports of the machine by screws through their centers at the outer sides of the upright supports, or better, directly in the same axle of the disks, at the inner sides of the supports. As the centers of the neutralizer bars are neutral, it is not important if they are electrically connected to the structure of the machine or insulated from it.
The basic Wimshurst machine is now ready. In a dry day, with the disks very clean, turning the crank shall rapidly charge the disks to a very high voltage, what you can easily recognize by the noise of small sparks between the sectors, the ozone smell, the electric field pulling the hair of your hands, and the effort you must apply to the crank to keep the disks turning. In humid weather, a hair dryer can be used to dry the machine and make it work. Some carnauba wax in the disks helps to make them highly insulating and nonhygroscopic.
To complete the machine, build the charge collectors. The disks become charged at opposite polarities at two quadrants, and at identical polarities at the two others. Adjust the position of the neutralizing bars to position the quadrants where both disks have identical polarities at the two sides of the machine. In the correct position, the disks pass first under a charge collector, and then under the closest neutralizer brush (easy to verify, as nothing will be collected if you arrange the neutralizing bars in the wrong position). The collectors can be two thick solid wires with U forms with sharp points directed at the disk sectors (not touching them, of course) fixed to them. The charge collectors are connected to a spark gap that is the machine output, and the assemblies are mounted in long insulator supports. The insulation of the charge collector and terminal assemblies is of fundamental importance. The insulators must be as long as possible and made of material with extremely high electrical resistivity, as acrylic plastic. Wood or similar materials act as total short circuits, and are not suitable for this purpose. The assemblies must also be kept away from any other part of the machine structure. I mounted the collectors and spark gap in a acrylic bar fixed to the machine support, with some screws to allow adjustments in the position of the collectors. Be careful to do not allow the collectors to touch the disks. This is the most common problem with these machines, and may cause extensive damage to the disks. It's recommended to make the collector points with flexible thin wire, so they can't scratch the disks. Do not leave any sharp point or corner in the assembly, with the exception of the charge collector points, or charge will be lost to the air. I made the spark gap and collector assembly with brass wires 3 mm thick, connected through aluminum balls. Screws cut at the end of bars with loops at the other end fix the spark gap bars at the chosen angle. The spark gap was made with two aluminum balls with 1 cm diameter, turned in a lathe. The diameter of the terminal balls shall be consistent with the size of the machine. Too small spheres result in weak short sparks and just corona if the terminals are separated beyond a certain distance. Too large spheres result in short, strong sparks and nothing if the balls are too separated, due to insufficient voltage. A good rule is that the diameter of the terminal spheres shall be of about 1/15 of the diameter of the disks.