What should a pilot flying IFR in IMC and on final, do if the vacuum system fails?
Your scenario isn't really realistic, the turn coordinator and the AI are almost always on different sources of power to protect against this very event. The TC is almost always electrically driven because it gives an alternate source of information. 

A loss of an instrument or even a whole suite of instruments is something instrument rated pilots train for, and should practice once they have their rating. We call it partial panel, where you use your other instruments to make up for the loss. If you lose your vacuum driven instruments you fall back on your electrical and non powered instruments like the altimeter, airspeed gauge, VSI and magnetic compass. 

Addressing your question, if I experienced a loss of vacuum on short final when I was established on an instrument approach I would most likely simply continue the approach and land. A loss of vacuum doesn't immediately cause those instruments to keel over and die, they keep working for awhile, getting less accurate as they spin down. Heck, if I'm on a short final I may not even recognize the failure! 

If I'm on a longer final and I lose my vacuum powered instruments I would also continue the approach unless there was alternative with better conditions or less hazards nearby. Your vacuum instruments aren't going to fix themselves, so there's no point in flying around waiting for something else to go wrong. 

The exact situation you describe in your question would be extremely unusual to say the least, this would most likely be caused by something pretty catastrophic that would mean continuing the approach is impossible. If I had a total powered instrument failure but somehow my ILS and radio stayed up I would continue the approach using the residual momentum of the gyros to keep situational awareness as long as possible, then fall back to my working instruments. I'd also declare an emergency, and ask for ATC to give me surveillance radar data, they can give me a running commentary of which direction I'm turning. 

Of course, if it was me I'd activate emergency AI app I have on my tablet GPS or my phone. Both use the phone/tablet sensors to produce a pretty reliable picture.