Sounds like your switch is physically damaged and will have to be replaced to work properly.
You can get a new switch here:
http://www.consolezombie.com/store/dsi-lr-...c17cdb000b3a9e5
Be advised that it is not a trivial repair and requires you to tear your DSi down to access that component.
Check out your warranty (which may exclude accidental damage) and make your choice.
1) Deal with a dead button
2) Attempt to repair it yourself with a new switch
3) Send it in for warranty repair and hope for the best
4) Find a repair center you can trust
5) Sell the unit AS-IS for whatever you can get and purchase a new one
The lesson to be learned here is that dropping any handheld can be very expensive.