Lenovo T61 Battery life 10 min?

A fast check to determine whether you have a problem with the power adapter, power cord or the laptop's battery is to turn off the laptop, remove the battery, connect the power adapter, turn on the laptop & see if it works.
If the laptop works directly from the power adapter with the battery removed, then the power adapter is good. In this case we can suspect that the battery is defective.

If it only lasts you 10minutes, it definitely means that the battery has reach its EOL (End Of Life). There is nothing you can do and you will need a replacement. If you are looking for the cheapest replacement, ill recommend eBay. Buy from a seller that offers 12months warranty. Prices may range from $30 to $50 Brand New.