I too went through this particularly annoying process. The solution that worked for me was to first go to:
office.com/setup
This forwards you to the point where you can enter the key that you have, and that you want to apply to your office product. It doesn't matter if the key that you enter is already part
of your account. What entering your key does is it takes you to your list of licences. Alternatively if you know that the product key that you wish to use is already part of your account list, then you can go directly to "Install from office.com/MyAccount",
towards the bottom of the key entry form.
Enter your account login credentials as appropriate. Now the annoying thing here is that if you've already added the key to your account, it is impossible to work out which one is which!
Microsoft - Sort this out!!!!!!
The best advice that I can give here is to look at the date at which you added a specific licence. It might also be possible if you enter the key initially that the correct 'install'
is highlighted for that key.
Allow the setup script to download and install and it rectifies the key settings for you.
Another option that I tried and worked on Win 8 was:
C:>cscript "C:\Program Files (x86)\Microsoft Office\Office
15\OSPP.VBS"
..and mess with the correct options. You need /inpkey to change the key settings.
Hopefully this saves you all some frustration, and stops you going bald like me!
Accurate as at June 2016