The battery can't be recharged

You may have a bad control board, so I'd recommend calling Dyson and making use of the warranty. Depending on the serial number, you should have no issues claiming warranty, even if this was purchased second hand. That being said, if you are looking to repair the issue, you must first be sure it is the battery and not some other part.

You can remove the battery and put it on a charger to see, then measure the voltage. I'd make double sure there are no shorts in the board, that should be relatively simple with the help of a voltmeter on continuity setting. Past that, I'd look at the board and see if it does provide power. One thing I have noticed with the cordless Dysons is that some work intermittently based on how dirty the filter gets or if the dirt receptacle is full. That in itself is not a fix, but also make sure the pressure sensor isn't shorted on your unit. I hope you can make use of the warranty, as this is a lot of work which may still lead you nowhere. Best of luck!

I picked up a v8 at the flea market for $25 with the same issue. Pried the battery apart and had the issue I suspected: A wire had been broken because it was pinched between the plastic body and a screw head where gravity pulls down on the vacuum when placed on the wall mount. Over time this disconnects the charging port from the control circuitry.

CAUTION: Taking the battery apart can be dangerous. There are exposed electrical connections everywhere once the battery cover is removed. None of these voltages can hurt you, but if you short out the battery cells (for a decent period of time) you can cause the cells to violently degas and catch fire. Also, the plastic cover is hard to disassemble without getting marred up. If you know of someone who fixes cell phone screens, they can most likely do this fix for you as it involves a lot of the same skillsets.