Shower Room Floor runoff

We just moved into a house with a shower room/wet room that was done a few years ago. We don't know the exact tiles used on the floor.

It seems that the floor in one area has dropped - just immediately outside the shower "tray", but still in an area that gets wet during use so it doesn't drain off correctly - it's probably only a matter of 1 - 1.5cm but is a pain. The grouting needs redoing anyway (esp to keep the water from getting under the tiles) - how might we best (and not super-expensively) restore a slope? Would we get the old tiles out intact? If not presumably we should tile over them but want to avoid a lip/trip if possible. We would be prepared to replace the current partial screen with one that divides the room with a sliding glass door...

Any suggestions welcome!

T

Laying new tiling from scratch is a satisfying DIY job if done properly; correcting badly laid tiling by somebody else, in my experience, is a job for a tiler. Why not at lease get one in for an estimate, it may cost you less than you think and - at the very least - you'll get some valuable advice.