Linear has a good selection of lion/lipo chargers too, probably a bit more of an "up to date" catalog than maxim in that regard.
the power pack that came with your laptop IS NOT a lithium ion battery charger. It is an external bulk power supply, and the charger is built into the computer. Trying to recharge a laptop pack with a simple charger will be quite a challenge. the laptop pack is a series + parallel arrangement, along with fuel-gauge chips (columb counters), charge balancers, anti-counterfeit id, thermal sensors, over charge and discharge protection circuits, etc. even if the old laptop is destroyed to where it won't run, will it still charge the battery? that will save you much work.