A freshly charged, good battery, should be more like 13.5v, so your battery is suspect. when you have given it a good charge, remove it from the charger and check the voltage every so often over a period of time. You should find there may be an initial slight drop, but after that voltage should be maintained for weeks. if it drops over a short time. The battery has had it.
When it goes back on the van, there should be an obvious increase when the van is powered up. This shows the van's power supply is working.
John