Leveling two different floor types
Basically you have to bring the level of the concrete up 1½".  1½" happens to be the thickness of 2× dimensional lumber, so you could take some 2×12's, lay that on top of the concrete, and you'd be at the extant kitchen floor level.  2 layers of ¾" T&amp;G subfloor should serve the same purpose, but, since ¾" plywood isn't actually ¾" thick, you'd likely need to buy ¼" luan or underlayment to make the levels match.  Ordinarily luan is bad for tiling, but since you'll be using floor leveler you can probably get away with it.

Once that's done, remove the vinyl from the kitchen, then use floor leveling compound on the whole space.  Once that dries you should have a nice level floor ready for tiling