As Big Chas said, you can't record the computer audio directly with the other (if you mean you are recording straight to the computer), but there might be some other ways you could get the video to have both the song and drums.
First off, I don't know exactly what you are planning on doing with your video. It might be for other purposes etc. so forgive me for the the following - but I thought I'd just say the other issue involved - If it's intended for youtube etc. it might not be copyright allowable) - sometimes however, bands etc. don't mind people doing tributes/instructionals etc. and they even link to them so others can see them. It might be worth writing to their publishing/record company and letting them know what you would like to do. They might really like it. I feel a bit bad saying that because I don't know what you will do with it. I just wanted to say it just in case.
Concerning how to go about doing it - it depends on how it's being recorded.
If you are taking the outputs from the ux2 and plugging it into a line input on the camera (you probably aren't doing it this way as you said you can't get the song audio on the video) then it will work fine.
If you are filming straight to a computer, then you could; get a digital audio workstation program (DAW) and put the song into it, then (possibly -I'm not sure if this will work) you can have the live drums being recorded in the DAW, and have the program 'resend' the combined audio out to something like virtualdub or a video program that accepts such sending programs.
There is a program called Jack Router which allows two programs to have the same thing at the same time i.e. audio from one program being sent live into another program. I haven't got Jack Router to work with my DAW. I have used it ( it may have been another program?) with other things (video) though, so it might be worth trying.
The video program would then record from the jack router input and take the video directly from the camera's.
Another way is to record your playing into a DAW and put a saved version mixed with the song, into the video.
To do this you would first put the song into the daw and record your drums as you play along to it. Then you could make an audio file of the resulting mix (of the song and your drums) and input it into the video. This will naturally result in slight sync problems but things like virtualdub can give you options to try to keep the video and the audio in sync. It won't be 'absolutely perfect, but it is a way it is often done. Audio recording sample rates are different to video rates, so there is a mismatch. For a song though it's not so bad as a long video where the audio can get really out of sync.
Hope this helps.