I think a good solder sucker or desoldering station would be a better investment than a tool specifically for LEDs. With my desoldering station, I can melt the solder and suck it off the contacts, eliminating the issue where one contact will resolidify before the second pad melts. To resolder, if it's an LED near the edge of the board I can hold my solder in one hand, hold the LED into the switch with the same hand, and use the iron with the other. For other LEDs, I would need to be a bit more creative, but it's still doable.
The reason I recommend a desoldering tool is it's significantly more versatile than just LEDs, you can use them for pretty much anything (join size permitting, of course). With my setup, I don't really have any more issues with LEDs than I do with switches.