You could download and install just the Windows Media Player, as it is free.
You could download and use a third party player like Foobar which would allow you to play virtually any common lossy or lossless audio format. And is also free.
Have you also gone in and checked your file associations to determine if the file format is non-associated with a program that could play it?
Search Windows for 'file associations'
Click the choice that says 'Change the file type associated with a file extension' or similar
Find your file type
Click 'Change programs...'
Determine if there are any media players on your computer that are actually capable of playing the file format. Usually very visible.
Per a discussion on the Foobar Forums, in order for Foobar to play .wma files, it does so by accessing the windows API (Application Programming Interface). If WMP or the Windows Media Format Runtime isn't installed, then it doesn't have the means to translate the file format. This is a contradiction of the Foobar Wiki, under supported formats, which says it can play .wma natively.
A workaround is the option of trying to install an older version of the WMP software by using Compatibility Mode to install it in Windows 8.1. This would provide the media formats for Foobar to reference, without having to install all of the extraneous material from the Windows Media Feature Pack.
Other options would be using other media players like Media Player Classic, which wasyl has already found will support the format.
Also, as suggested by Ramhound, using the media player VLC.