No battery is detected

you need a new battery. thats the only answer. the fact it only lasts 5 mins is a classic example of a dying battery. the fact its not detected is probably because each cell has dropped below 3.3v.

lithium ion batteries dont loose voltage in the same way as a Nimh or Nicad battery does, they maintain the same voltage until they are flat, and then go off slightly.
If the voltage drops too low it can do irraversable damage to the cells. and I think what has happened here is that the cells have dropped below the threshold and is no longer detected by the motherboard as its looking for more than 3.3v per cell.

See if these help:
How to improve your laptop battery's performance | Dell US
Dell PC Diagnostics | Dell US

The machine isn't offered by Dell anymore, so it could be a weak or dead battery - they only last a few years.
The Dell links above will help you determine the condition of your battery. If you need to replace it, you'll know for certain. If it's something else... you can rule out a bad battery.

A replacement battery is not that expensive, but if you don't need to replace it, don't.