How Can I Speed Up the Healing from IPL Burns?

This is a tough problem.  IPL is great for the face, neck, chest, arms, and hands.  The results on the back and legs are much less predictable and can result in unsightly pigmentation.  You may want to try bleaching cream on the areas surrounding the white squares to make them less noticeable.  If the white areas are still present one year after treatment, chances are that this is a permanent alteration.  The white areas can be tattooed with flesh colored pigment to try to match the surrounding area.

Sounds like the burns (blisters) were more than superficial 2nd degree burns. Any burn that blisters is a 2nd degree burn but it can be a superficial 2nd degree burn and heal uneventfully. But it can be a deeper 2nd degree burn and destroy the melanocyes moreso. This will result in loss of pigmentation of the skin later on. This can be permament. I would wait several months for several skin growth cycles to take place then evaluate. Trying to reduce the normal pigment surrounding the white areas with lasers or chemical peels may help to blend. I don't know if you can do anything to repigment the white areas.