LoL, OUCH I know the pain of breaking that tip, more than once too:lol:. In both cases I broke the tip near the original end or in the middle between the original end and the second eye. What I did was remove the "Tip Eye" and placed it onto the new end. That way I could still keep the shape of the Tip eye and a little bit of the length. That is of course you didn't break off near the second eye than just cutting it off there is fine, lol. As for sensitivity, for float fishing it doesn't bother me much since I use a bobber for indication and it doesn't seem to be effecting the fight with a fish. For drifting and spinning, I notice that there is a bit more stiffness when feeling for the bottom, but it isn't that bad.
For the bobber and jig setup you are asking about, you have the right idea. Try to tie the jig so it is presented horizontally. I usually give myself quite a bit of leader to work with depending on the depths. I change my jig type about every 15-20 minutes, so maybe after an hour and half it will be time to tie an entirely new leader. Good Luck!