Mice that are eating through the boards under kitchen cabinets

One way to avoid splintering the front of the drawer is to clamp a small piece of wood on top of the area to be drilled, and drill through that bit of wood, into the front of the drawer.

This should (if you have the clamp firm) prevent any sparring or splintering on the finished surface of the cabinet drawer.

Try it on an inconspicuous area first (like a drawer near the floor or ceiling) to be sure it will work.

Another option is to try using a brad point drill bit (like one of these). They usually create a cleaner hole than standard bits.

First off you will have to clean off the grime. Any attempt to dress up the surface of the finish with the grime still present will look worse in my experience.

Cleaning the grime off can be attempted using a washing solution of TSP and hot water. TSP should be available at your hardware store in powered form as TSP or trisodium phosphate. It is an awesome cleaner but will require you to use good rubber gloves that cover up your lower arms.

After you get the grime removed you will be in a position to evaluate the condition of the finish. It the finish is in relatively uniform condition and not worn completely away or cracked / peeling you may be able to improve the looks a lot with an nice quality oil wipe. I would not recommend Pledge type products for reconditioning the finish surface.

If the finish is mildly non-uniform but clean you should be able to pipe on a thin coat of wipe on combo stain/finish. Select a color very close to the current appearance for best results. Make sure to try this out on a non-obvious surface to make fully sure that you are satisfied with the results - in other words do not rely solely on my advice and do read and follow manufacturer instructions.

In cases where the finish is completely gone and bare wood is showing it may be necessary to go to the extreme and strip off all the old finish, sand and re-finish. This can be a huge project (I can speak to it after doing a whole kitchen some 25 years ago and many cans of very smelly strong stripper later) but can be the best way to get cabinets look close to new again. Mine in that project had two coats of paint over original baked on blond finish on ash wood. After re-finishing they looked awesome.

If there is physical damage to doors and drawer fronts you could consider another option that is available. There are businesses that will apply a new covering all over the visible parts of the cabinets. I have a hard time recommending this approach because I think the results of such effort are ugly.