First off, tell the girl that you need some space. Your dad is dying and you're starting a new job and it's getting to you, she should understand (and if she doesn't, well, screw her). That's 1 problem solved!
Secondly, sorry to hear about your dad. Unfortunately, there's really nothing you can do about the stress and sadness. Have any of your friends/family had a parent die? It's a touchy subject, of course, but you could ask them how they dealt with it. Seeing someone who has gone through this and come out the other side should give you hope.
Lastly, try to remember that your dad wouldn't want you to be stressed out and not doing your job to your fullest because of him. If he's anything like most dads, he'd rather you were acing your induction and then coming back to see him!
Good luck.