It looks like the inductor is used as a EMI filter ("pi" filter with the two adjacent capacitors) on the battery connector to filter the charger's switching converter noise. The value shouldn't be so critical, and the laptop should even work with you just bridging it with a bit of wire.
It is connected to a BQ23748 charger which doesn't seem to have a publicly available datasheet. However, the BQ23745 suggests a 4.7uF LC filter. Having two in parallel, means that 8.0 uH is about what they suggest for this different part.
If one of PL9 and PL10 is fried, then probably both are bad, preventing the laptop from running off of battery properly. If the battery is removed and the AC adapter connected, does it work properly?
If you remove PC206, then you could measure the value of the two parallel inductors on the board which are next to the AC adapter input. If your LCR meter has a guard, you could also connect the "Guard" to your board's ground and measure the inductance across the inductors without removing any components (at low-ish frequencies), if the guard terminal is able to overpower the capacitors.