A smith might approach a repair thus:
(1) grind out as much rust, dirt and misc. junk as possible
(2) hammer-weld the split (the success of which will depend on the skill of the smith and the exact steel used in the making of the axe) reforming as much of the mushrooming as possible at the same time
(3) reform the eye using a punch or drift hoping that the weld holds (punch may need to be made or adapted to suit)
(4) clean up, regrinding, heat treat (hoping the axe doesn't go 'tink' on quench), sharpening
Time: perhaps 1-2 hours considering it'd need babying. Shop-rate typicaly £20+ per hour. Economical, no. Interesting, yes.
Of course it might be plugged with an arc-welder and a suitably shaped plug. Lots of cleaning, pre-heat and post-heat with a torch, possible re-heat-treatment and lots of time messing about with files trying to get the eye right again.
If it were mine I'd clean it up and use it as a talking point in the workshop, at meets, down pub... (okay perhaps not last one.)