From this one photo, looks like there is quite the transition (i.e. lots of room for buildup). I would not bother trying to remove the tiles if there is no need to.

Personally, I would put a layer of Schluter Ditra underlayment (use polymer modified thinset), let this cure. And then lay some new tiles over this. With this underlayment, you can lay large tiles if you want (personal preference) and not worry about cracking from the house settling/moving from temp changes) as this is the role of the underlayment (movement absorber,best way to describe it). Non-modified thinset OK here, use the correct trowel notching for the tile size, and backbutter the tiles before setting. Let cure and then grout the gaps with whatever color you choose (lots of options to suit the decor/tile color).