Identify the pins of the circuitry that resides within a laptop battery
It's just four 18650 cells connected in series with a PCB consisting
  of a few ICs


Yes, the problem is likely in those just little few ICs and/or power switches. Laptop battery is a fairly sophisticated charging and cell-balancing system, with a bunch of protective and service circuitry, including charge monitor, all based on embedded microprocessors. It is called "smart battery". If the battery communication with host was successful, it means that some power element/fuse/MOSFET has died/fried/burned.  

For ASUS-made battery (as well as for the vast majority of them, including Dell/HP etc.), the components are Taiwan/China/Japan sourced, an it is practically impossible to find replacement components, and if the MCU controller has died, then it is a dead end. If you are extremely curious, you should reverse engineer the PCB schematics and post it here, maybe then people can give you some practical advice.