I'm more worried about the cooking than the workout space. Please be sure your Mom is safe. Lately there have been a lot of garage fires around here as families try using charcoal or propane burners in the garage. Without a vent hood or floor, your Mom is asking for trouble. I know in many places a spice kitchen is used to keep smells and grease out of the house while cooking, but a garage is not a spice kitchen. Do you rent or own? Is there an opportunity to properly convert the cooking space?
For your workout area, I would rent a steam power cleaner, and blast the whole floor on a warm day, then leave the garage open to dry out for a few hours. You can get cleaning additives to start with degreasing (I like simple Green) but the water pressure will do a lot of the work to remove the problem.
Once the garage is clean and dry, buy a roll of subfloor pad and a box of laminate flooring and make a pad big enough for your exercise area. The Laminate should be the cheapest you can find - around here its about $0.75/sf. You just want something to get you off the floor. The padding beneath will insulate and cushion.
In the cooking area, I would consider a rubber mat to catch the spills - something you can drag outside to scrub down when it gets gross. Alternatively, consider a sheet metal tray that the roaster/burner can sit in to catch the grease. You may be able to buy the steel and manufacture this yourself.