If she likes music, I would suggest going with some Kinect rhythm games for now. It's something they can jump around, and flail to make it seem like they're doing great, even if the stage keeps giving them 0 points, but at least they think they're playing along, lol.
There are some other Kinect games she may enjoy, but some of them will require you to kind of help move things along, or for you to play through first to progress the game, and open up more options.
I mean, handhelds are great if they're a bit older, but to be honest, most games still require some reading, and motor skills aside, I'm not sure how many of the handheld games are that fun for a 3-4 year old if there's any story in it.
If you have a decent tablet, I would consider possibly starting with some simple app games. There are tons of app games that are built for tablets now, most of which are free, and great for kids. My niece just loves those games on the tablet, and can just sit there playing them for awhile. It's my brother in laws master plan to transition them into a handheld when she's a bit older, but for now, that seems to be pretty effective, and the apps are mostly free.
Of course, the tablet is much more expensive than a DSi (in most cases), so depending on how rough or gentle she is with it, it could end up in the trash, lol. Just make sure you get a good case. ;P
But yeah, Kinect games are probably a good way to go for now, and you can participate too at times if you want.