@Leon Horman
For some security reasons, the complete key cannot be obtained on your computer.
What we can do is to get the last five keys through command in Command Prompt.
If you have a valid proof of purchase of this Microsoft Project 2013, I suggest you try to contact Microsoft support online directly to confirm whether they can replace the key for you or provide other options.
More information about the key, you may refer to "Using product keys with Office".
Hope the information could be helpful.
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