stuck half shaft

I had a half shaft go a while ago. Just got round to fixing it. After removing the broken one it's clear the broken end is in the diff. Fast forward many swear words and grazed knuckles and I have the diff on the work bench but for the life of me I cant remove the broken piece. It can move about 8mm in the housing but wont actually come out. Any ideas would be great feel free to take the **** if I've monged it and missed something obvious.

Possibilities might be 1) drill and tap a thread, wind in bolt through a cover plate to extract the bit, 2) drill and use an ezi-out, 3) if you can get to it, weld a rod onto the end and pull out, 4) dismantle the diff so you can extract through the other side.
Options 1-2 may well put swarf into the unit which you don't want (drill slowly, vacuum cleaner nozzle might stop this), option 3 you may not have space for, option 4 is probably where you may well go with this thing though. I've never had to do one, but these are the options I might consider. Someone will come along and tell you and me that the shaft is fully hard, so options 1 & 2 are defo out, even with HSSCo bits.

You should be able to poke it out from the other side

It moves but only a bit then gets stuck fast. I've tried knocking it back and forth but cant seem to get any more than 8mm of travel. Thanks for the quick replies though guys. I'm thinking option 4 is going to be the way to go on this.

Finally got it out. Again thanks for the advice guys. It was option 4 with the addition of a bfh.