Hi James
I have had the exact same problem with my SP3 over the last week. Updated using the latest firmware and because of that the type cover is no longer working.
Have tested different type covers from other SP3 and SP2017 on the non working SP3 with the same result yet they work flawlessly on their own machines. The type cover from the faulty SP3 works flawlessly on the other two Surfaces.
Have over the last few (sleepless) nights:
Reinstalled with different versions of Win10 (1709, 1803)
Disabled and reenabled drivers in device manager
Tried the 2 button reset (Volume up and power button)
checked for updates
Downloaded and ran (twice) the diagnostic tool
Created recovery image from MS website which installs 1511 for it to then update itself.
I was on the telephone yesterday with a MS Surface support technician who ran me back through the bottom four of the above methods to no avail.
I was then told my only recourse to get a working machine was to buy a new one for around $500 (why dollars? I live in the UK!) which works out to around 600 UK pounds! I have to pay for MS breaking my computer! Now that's a business plan!
This problem was caused by the latest firmware Bugs that MS pushed out to everyone and they should be fixing it. They have already acknowledged that they bungled the SP4 update (see Surface Pro 4 - broken after 26th July update) and will be fixing it.
Will MS be issuing a fix for their botched SP3 firmware update too? The problem is not with Windows. It is with the Firmware/EUFI. How do I know this? because the keyboard does not work when booting into the EUFI setup whereas on a working SP3 it does.
MS needs to fix this as they broke it. I do not want to be told its "tough luck" and "You'll have to go buy a new one" by the MS support technician!