Hand Burn

So I burnt my hand last night on the metal handle of the frying pan that was in the oven; probably one of the worst pains I've felt.

I soaked it in water for an hour before finally being able to take it out. The pain was unbearable.

I ended up vomiting, feeling lightheaded and sick the rest of the night.

I then applied polysporin for burns, non-stick pads and gauze to the area and eventually went to bed.

Fast forward to today, the pain has lessened and the blisters aren't as bad as I expected them to be but I still feel nautious, my hands are shaking, and I've been shivering all morning.

So my question is, is it normal to feel sick like this the day after? How long do you think this will last?

You are suffering from shock. If this doesnt subside after 24 hours get medical attention. 

It's shock. Shock is a medical emergency. It is concerning that it has not subsided on its own (when I broke my fibula hiking, mine subsided after about 45 minutes; at which point I was able to get up and slowly hike out). I would urge you to get someone to take you to the ER/urgent care and get checked out. 

Sounds like shock to me.  Are you taking anything for the pain?  If not I would take something, bundle up, do some deep breathing and try to relax and perhaps have a little nap.  If you are still feeling bad after a period of time give your doctors office a call and see what they suggest since you have done a good job caring for your burns.  I did the same thing as you with a cast iron fry pan in the oven.  I was not as burned but certainly felt dumb.