Construction adhesive may hold, but it also may release over time due to moisture exposure at floor level. You might want to consider a high grade silicone adhesive, instead. Regardless of what you use, you'll still end up staining the grout (a consideration only if you ever intend to take the island away). I believe that granite counter installers routinely depend on silicone and the weight of the granite to keep countertops in place. If you try adhesive and it fails at some future point, then I would mount a hidden frame of 2X4's to the floor and attach the cabinets to that. I would do this by drilling through the tiles. A hammer drill and the right bit will make short work of it. Then you can fasten the frame down with the something like tapcon screws. You could even add some silicone adhesive: this glue and screw combo should be plenty strong. As for powder actuated nail drivers, they take some skill to use correctly and get the depth correct, and are very dangerous to use without safety training: they're basically guns that fire nails instead of bullets. I'd discourage this approach.

The tiles cracked because the subfloor supporting them flexed. There is nothing you can do to the mortar bed or drypack to fix this. The crack is not important. The subfloor around that column needs to be reinforced.

Because it seemed important to know how bad the situation is, I removed the tiles and mudboard in the problem area. Underneath is just the sheathing. You can see that most of the wood is still in tact, but it's definitely rotted in the corner by the tub. That spot is still very wet, smells earthy, and I can dig into it with my finger. In a 4" x 4" area, I could make a little 1/2 inch hole easily. Now I'm wondering if the best solution might be to: 1) wait for wood sheathing to dry; 2) put down some kind of underlayer or mortar, and then 3) just put on some of that Flex Seal - as a temporary solution. Or, should I put down some kind of PVC board, or something like that over the sheathing - and then put mortar on that - just in case there's reason to worry about the sheathing in that corner giving in. and the PVC board would keep us from falling through the ceiling (until we do the full renovation)! Thanks everyone for your help!

thanks mainenell. Any suggestions on the type of caulking, Im sure once I go to Lowes or HD, there will be so many types and brands with minor and major differences, I will get lost. Most frustrating thing about a DIY project is to pick an item at these big box stores. Is this the current one grout?