First of all assume that your KMS is functional, having Office 2013 key installed. Also the client is able to reach the KMS server on port 1688, and the setting in your internal DNS system is correct.
You can check registry on the client for this:
Here are the two commands you can try specify the KMS server and do manual activation.
cscript ospp.vbs /sethst:fqdn_of_your_kms_server
cscript ospp.vbs /act
For Windows activation, the commands are like this:
- cscript slmgr.vbs /skms fqdn_of_your_kms_server
- cscript slmgr.vbs /ato
Further, you may want to try uninstalling the product key on a client and then reinstalling it.
slmgr /upk xxxxx-xxxxx-xxxxx-xxxxx-xxxxx
then slmgr /ipk xxxxx-xxxxx-xxxxx-xxxxx-xxxxx
Reference page from Microsoft: http://technet.microsoft.com/en-us/library/ee624355(v=office.15).aspx