Wow, you're lucky the LED didn't fry. The solder connections are the only way to remove heat from the thing. While a little pressure from the reflector or something can provide an okay electrical connection, it provides a terrible thermal connection. The light must not be pushing the emitter anywhere close to its limits.
If you bought some solder paste and plan to re-solder the connection, make sure to clean off the old solder as well as you can, and definitely clean any flux residue before applying the paste. Anything that looks like it could be removed should be removed.
Isopropyl alcohol (IPA) cleans most fluxes pretty well. The higher the purity the faster it cleans and the faster it dries, but if you are only doing one it probably doesn't matter. Even 50% works, though 70% and often 90% are available at drug stores, even dollar stores.
We use technical grade IPA at work (I think it's essentially 100%, but I never looked it up), but it's most readily available in 5 gallons and larger sizes. I use denatured alcohol at home because it's cheap, available by the quart or gallon at the home store, and I have it on hand for working with shellac. Plus, it works even better than technical IPA. Now that I'm thinking about it, I don't know why we don't use it at work. I should look into that. Acetone works even better, but due to its toxicity (low, but way higher than alcohol), I prefer to use it as little as possible.