How do I repair this tile without removing it
You don't repair it, you patch it, and don't use the toothpaste. I have seen repairs done with a small bead of caulk. You can rub it in good and then clean up the tile. This looks like hell. You could also use some white or clear acrylic nail polish and carefully go over the cracks. The problem is someone probably cracked that tile by kicking it and it could crack again. The tile might not be supported properly.
I know you said you didn't have any spare tiles but this is a pretty common tile so take some measurements and head out to a tile warehouse and look for a match and fix this right. Also, the grout is missing on the lower left corner and I would think there should be a caulk bead between the wall and floor.