How to find which Office license key was installed

Office like most of Microsoft products, comes in an OEM version and a retail version and an enterprise version. The key from one version can't work in another.

First question - ask your client - did Office come with this machine when they purchased it? If so, most likely it was a OEM version and you have an OEM key. You need to get the OEM iso. If it was preloaded, see if they have the original disks. Check the casing for a COA sticker.

Second question - did they install it after? If so, again ask for the disk. See if any other computers at the same time were loaded - if so, may be same version.

If this is a large company, they may have a enterprise version - if so they usually have a special key and special install disks or downloads.

The standard way of doing this is just uninstalling the copy of Office from the current computer and—because Microsoft only gets notified of new installs/activations, not uninstalls/deactivations—then using phone activation when you've hit the activation limit permitted by your license. If you explain the situation to them, they should give you a new key to activate Office on your new computer.

See this Technet question and this Superuser question.