Hi David,
I am Dave, an Independent Advisor, I will help you with this.
The Tablet Mode button will not appear on the Notification panel if Windows does not sense a touch screen, you can only turn on Tablet Mode from Settings - System - Tablet.
With regards to the touchscreen not working there are may reasons for that, did the service center actually install a touch screen or a normal screen, also if the tape connector going from the touch screen to the digitizer panel is not connected correctly, then the touch screen will not work.
Basically, you need to take your laptop back to the service center and have them get that touchscreen working for you, because most likely, it is their fault it is not working.