It sounds to me as though the contacts in the motor are just about
shot. What you're smelling is ozone - a really good reason not to
run the drill in an enclosed space for long. The best thing you can
do to salvage this equipment is to contact the manufacturer and find
out A: if the contacts are replaceable, and B: if they still
manufacture the necessary contacts or have any in stock. If you're
not up to the job of replacing the contacts (it's fiddly and
frustrating, but if you weren't into that you wouldn't need a
dentist's drill), they might be willing to replace them for a fee.
Make sure you have the model number handy when you call.