Tips:
1) If screwing into MDF.. be sure you predrill pilot holes or you will end up splitting the MDF weaking it.
2) If you are routering out from the bottom and pusing the stick upwards, I would suggest buying a caulk-tube size of what's called "Subfloor Adhesive", and the numbers on it around here are "PL400". This stuff is amazing. I used it to glue down 3/8" plywood in the rafters of my garage, and within an hour of applying, you could not remove the plywood from the 2x4's without a crowbar. After a few weeks/months, it literallly is stronger than the glue used in the plywood and the plywood itself. A good amount of that between the top of the joystick base and the underside of the wood (ensuring it doesn't ooze into the shaft/open area), would secure the joystick without needing any other attachments, so long as you let it dry long enough.
Of course it makes it a pain to ever remove, but I'd use some of this in addition to your screws, and also use in the pilot holes before you screw it in place.
This is not the same as a bottle of elmers wood glue. This stuff comes in caulk-like tubes and is very thick and dries very fast. They have separate levels for gluing styrofoam to concrete (PL300), paneling to walls (PL200), and this stuff (PL400) that is the strongest. It's about $1.69 per tube. Heres a link to it:
http://www.stickwithpl.com/PL/PL_products/PL400.htmIt's so strong I'd say you could safely hang an entire sheetrock ceiling without a single screw/nail... so long as you could cure it in place long enough. However they actually make a PL100 glue for allowing you to hang sheetrock with fewer nails.