photo of sink and nails (please pay no mind to the bathroom filth my tenants left behind. Townhome hasn't been deep cleaned yet)

The tenants wrecked my apartment. I'm not sure how this pedestal sink became detached from the wall.

I found spike nails on the floor next to the bathroom when I entered the townhome after my tenants left. My dad told me that these nails were used to fasten the sink to the wall. He then used the nails to fasten the sink to the wall again. The left side nail is currently installed at the moment but the right is not. Even when both nails are installed, the sink is still not firmly fastened to the wall - the sink moves a little but only if you purposely try to jostle it...

My dad suggested that the nails should first be coated with some kind of caulking/filler to help fill the hole that that nail was originally driven into, to help fit the nails firmly into their holes.

Would this be a good way of keeping those nails firmly in place so that the sink is stable? How would a pro handle this?