Lydia, there really will not be much of a long term fix to burned Formica. If you sanded the area well, the paint might stick for a while. What ever you do, please remember that this is a food preperation area. Use products that are nontoxic, should they flake or chip off--and they will eventually.
If the Formica pattern is still available, the next best solution is to have someone who is handy with a router take out the burned section and replace it with a "dutchman".