RE: Catch-22: No BIOS updated without battery, no battery life without BIOS update...
Updating the BIOS will not fix the problem.
Check setup (F2 at powerup). If the AC adapter shows "unknown",
First, try a new Dell OEM AC adapter. If that doesn't solve the problem,
Second, replace (or have a shop replace) the DC jack.
If neither of those produces a positive result, replace the mainboard.
This is not a BIOS issue - it's a hardware issue with the AC adapter, DC jack or mainboard.