Try removing the battery and powering up with just the AC adapter
From the 5310 service manual which I believe is similar to the 5315 and has been uploaded and linked on the 5315 device page
Power System Check
To verify the symptom of the problem, power on the computer using each of the following power sources:
1. Remove the battery pack.
2. Connect the power adapter and check that power is supplied.
3. Disconnect the power adapter and install the charged battery pack; then check that power is supplied by the battery pack.
Attached are a couple of excerpts from the manual that may be useful. As they are from the 5310 manual the voltages may not be accurate.
May also be worth checking the connections onto the main board - I have highlighted in YELLOW the battery connector
BTW If I am incorrect about the similarities between the 5315 and 5310 please let me know and I'll correct the information accordingly.