If you recovery program applies a .wim as they usually do - you can't get the key directly from that .wim file.
You could try marking the recovery partition ACTIVE - use this free boot cd http://www.partitionwizard.com/parti...otable-cd.html
You might boot straight into the recovery program.
If that doesn't work, you might be able to apply the wim manually, but you would need to create boot media to do that - and it would take weeks to show you how.
Then the best thing for you is to order the recovery discs from your manufacturer - they don't usually charge for them - perhaps just postage costs.