Try getting in contact with Microsoft through chat. At first it will be virtual agent. However after answering a few questions you will be able to chat with live person who will be able to help with your activation.
You can perform a clean install (downloading windows 8.1 from Microsoft) and delete all partitions. You might need to disable secure boot in Bios and enable legacy boot. I like doing clean installs of operating system and delete all partitions, even recovery partition as I can create them myself by creating an image of the freshly installed OS with all drivers and software I choose. This image I keep for any future problems.
So, yes you can delete all partitions and install in unalocated space. You might not even need product key because the digital key should be in firmware of Bios and should automatically activate Windows.
Let me know how it works out and if you need any help,
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