How to repair fallen tile near kitchen floor?
To be honest, I'd probably be lazy here, and stick it back on with a thin film of a good commercial adhesive, thinly buttered on the back of the tile. Leave the old adhesive as it is. The old tile will fit perfectly. This is NOT something for superglue, but perhaps for something like liquid nails in a tube. Buy it in your local home center or a decent hardware store.

Use some tape (the handyman's secret weapon is duct tape, so says Red Green) to hold it in place, but any decent adhesive will have enough tack to keep a tile there while it dries.

Really the hardest part to do is to re-grout around the replaced tile. I'd not be at all surprised to find that available in a squeeze tube too. So I checked, and indeed it is, here, for example.