The simplest solution that I can think of is to first see if there is a way to insert screws into the base of the wheel. If this is possible, you will need a slab of timber that you can screw to the base of the wheel. Countersink the screws, with 20mm flat washers, so that the underside of the timber will remain flush, or flat is another way of putting it.
If there are no threaded screw holes available in the base of the wheel, remove the base and drill four holes and use zinc coated bolts, with washers to spread the load. Place the four bolts with washers on the inside, then a single nut on each bolt so that they cannot fall back inside when you re-assemble the wheel housing. You will have also marked your timber with the corresponding bolt holes, so once the timber base is complete, you can carefully lower the wheel housing with the four protruding bolts onto your new support base.
A piece of timber 25mm thick would be soild enough and the fixing nuts and washers could safely be countersunk into the timber and there would still be enough to make the bolt connection solid. A small countersink hole may also be required on the upper surface, this will allow room for the nuts fitted to the wheel housing.
When shaping the timber, allow 100mm on either side of the wheel body to allow you to use a sturdy g-clamp on either side.
For extra grip, glue a 3 or 4mm thick sheet of rubber, or a similar non-slip material to the underside of the timber. This will reduce vibration to the desk, and help with protecting the desk surface. On the underside of the desk, use small pads of ply where you fit the g-clamps, this will prevent placing circular marks into the timber, and it will spread the grip force from the clamps over a larger area.
The only drawback with this method is that the g-clamps must be large enough, within the G, to clamp securely enough to prevent the wheel and its base support from rocking on a front to rear axis. I would think that 100mm or more from the front edge of the desk into your base should be adequate. It may not look pretty, with a clamp on either side, but as a DIY'er of dare I say it, considerable skill, I can guarantee it will work.
Use a 25mm spade bit to create the countersink holes. Use hardwood, not pine. Sand and then paint the entire surface gloss black, then fit to the wheel. Measure twice, cut once. Make a template if you need to. Follow the shape of the existing wheel base. Good luck.
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