Your laptop is 11 years old so it is likely that the battery is dead but read on.
Here is the user manual for the Dell Precision M6600 laptop:
https://dl.dell.com/manuals/all-products/esuprt_laptop/esuprt_precision_mobile/precision-m6600_owner's%20manual_en-us.pdf
See page 131 - 133 for battery light codes. Could be a dead battery. Did you try taking out the battery and just use the charger?
Are you using a Dell charger? Non-Dell chargers ''could'' cause problems.
You could also try disconnecting the CMOS coin battery, as explained on page 49.
First disconnect the charger, remove the main battery, disconnect the coin battery (if you have a multi-meter check the voltage - 2.5 to 3VDC - if under that replace), press and hold power button for 30 seconds to discharge any power, re-connect connect coin battery, ''leave out main battery'', plug in charger and press power button. Does it turn on?