First of all: rule out thin-film resistors (should be intuitive, actually). Thick-film resistors can work well. But, as always, assume nothing: read the data sheet!
Wire-wound resistors have a good surge reputation, but even non-inductive resistors have higher inductance than, for instance carbon-composition types.
Carbon-composition is an old, but still reliable technology and should be considered, especially if cost is a concern.
If you can spend a bit more, ceramic composition might be considered. Heat tolerance is good.
Finally, don't forget cement resistors, which also have good heat tolerance.
You want to use exactly the same nichrome wire that is already there.
The trick is to find the mid point and run it as two resistors in parallel instead of the whole wire in series as it is now.
Think of what you have now as two pieces of nichrome wire in series. Each piece sees 110 V across it. Arrange them in parallel, and you can put 110 V across the whole thing and each piece won't be able to tell the difference.
Added
As Trevor pointed out in a comment, while the nichrome wire won't see a difference, the rest of the unit will see twice the current. You have to make sure that the line cord, switch, and everything else where the full current passes thru can handle it. You say this is a 700 W device, so will draw about 6½ A at 110 V.
Another issue is whatever is in the toaster that decides when the toast is done. There are various mechanisms for that, and most probably won't work right with half the intended voltage.
All around, if your aim is simply a toaster that runs on 110 V instead of 220 V, go buy one. If your in this for the adventure, then go ahead and experiment with what you've got. Don't expect great results, though.
Best Answer
Your premise is largely correct but there are a few things to take into consideration:
Mathematically, you are putting energy into the system and this can be calculated by \$ E = \int P dt \$. i.e., the area under the power-curve.