How can I raise a bonus room subfloor level with the adjacent floor?
2 1/4 inches height difference could be made up with 2 x 4's on their side screwed in and glued over the existing joists and 3/4 OSB sub-floor on top. That will be 1/32 lower than the abutting sub floor when you take into account the actual thickness of the lumber. 

You could use 2x2's, but often they're more expensive than 2x4s

Glue and screw everything to prevent squeaking.

The tricky bit will be the hardwood - carpet transition.  From your photo, which gets pretty pixelated at high zooms, it looks like you have 3/4 inch hardwood. If there is still a tongue on it, you can buy a reducer piece that will drop the height down 3/4 of an inch smoothly.  If it's a groove, you can buy a double tongue, which converts a groove to a tongue.  

Otherwise, you may wish to have a threshold specially made for this purpose.