FreeEnterprise, I also have this problem and am in a similar situation to you where the laptop is a throwaway anyway, so I want to attempt the re-solder even though I am a novice. I have my board disassembled and have examined the max 1987 chip, and am quite chocked at how small it is, to be having 48 connections. I am going out tomorrow to buy the tiniest soldering pencil (less than 20w) I can get as my soldering station is 60w and the tip far too big.
Even looking at the chip with my magnifying glass it still looks like brain surgery. From what I understand about your posts you said no additional solder is required, just to re-heat the existing solder. Each connection looks so tiny, how can I be sure I have 're-flowed' the joint?