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t0-ster
4.2mm is the usual tap drill size for M5×0.8, using a 4mm was always going to break the tap unless you were trying to put a thread in some cheddar. It's really hard to apply pure torque to a tap by hand and any eccentric force will easily snap a small one even if the pilot hole is the right size, so a drill press with a tap follower in the chuck would have helped, but not enough to get you past the problem of trying to tap Ti (as you have discovered, and as everybody else already knew, an absolute bastard to cut) in an undersized hole. If you ever get the remains of the tap out, you might as well go to 4.5mm on the pilot, it will reduce the cutting force considerably and having the minor diameter of the finished thread oversized will still leave sufficient strength in this application.
Bruce Whitham is the god of thread repair, you might learn something useful from his Getter Out playlist https://www.youtube.com/channel/UC6HG4kXPSgb1Bhx6CY9e6iw/playlists
The chemical method seems to be Ferric Chloride, which you should be able to buy from Maplin as it is a common etchant for PCBs. It dissolves steel but not Ti