You'll need your Mnemonic Phrase
Unfortunately, without your Mnemonic Phrase (or a backup of your private keys) there's no way to recover the accounts. So you'll need to get that phrase off your other computer at a convenient time.
Once you have the recovery phrase, you'll face another problem, how to enter it into the new MetaMask instance once it's initialised...
Resetting the vault.
Once you have initialised the MetaMask extension, there's no built-in way to reset the vault using your original seed phrase.
If you want to start from scratch, you'll need to uninstall the extension and reinstall it at which point it will ask you if you have your seed phrase available again.
This will however result in the new vault from being destroyed and replaced with the old one, which may or may not be desirable.
If you were using the Google Chrome browser in Windows 10, you should be able to find the location of the Vault by going to this folder location (you need to be able to see the hidden files):
C:\Users\USER_NAME\AppData\Local\Google\Chrome\User Data\Default\Local Extension Settings\nkbihfbeogaeaoehlefnkodbefgpgknn
In that folder you'll see a file called 000003 (or something like that), open that file with a text editor or code editor software like Atom (https://atom.io/).
In the text editor search for the word "vault" and copy this part that looks like the one pasted below:
{"vault":"{\"data\":\"wwpXXtFCqZkYsWfeEwItZjJ0Cc7mRVjG47Dqh+ztL1PiCG6Izhg+zG0mM+H2ykyjz3X0RNhAE6IVsWFZamcZ47B4sVi4SvUxrMhARm5L3yHPxr3UsyGrOXmthyVMgEGmjwlmnFCNd2nMZ2o8/sRMra8FupurqevnBv57FiYpEEs7gPpFHv6587aL44MmKD8Snv4JLFqiqmlK82Waq5F+Iv9mw2sFVAL9mgZBSgFgbWdB3TsKVB2k\",\"iv\":\"rkUQlNcGTxBE0My7a/bCXw==\",\"salt\":\"HcKyNfGzaRALRQ0DlKgcIe5Uk30iI/M//oG6w8vX8Nk=\"}"}
Remove the two parts in bold. Now use the Search and Replace feature of the editor to change all the \" to "
You would now have something like this:
{"data":"wwpXXtFCqZkYsWfeEwItZjJ0Cc7mRVjG47Dqh+ztL1PiCG6Izhg+zG0mM+H2ykyjz3X0RNhAE6IVsWFZamcZ47B4sVi4SvUxrMhARm5L3yHPxr3UsyGrOXmthyVMgEGmjwlmnFCNd2nMZ2o8/sRMra8FupurqevnBv57FiYpEEs7gPpFHv6587aL44MmKD8Snv4JLFqiqmlK82Waq5F+Iv9mw2sFVAL9mgZBSgFgbWdB3TsKVB2k","iv":"rkUQlNcGTxBE0My7a/bCXw==","salt":"HcKyNfGzaRALRQ0DlKgcIe5Uk30iI/M//oG6w8vX8Nk="}
You can now go to the Vault Decryptor https://metamask.github.io/vault-decryptor/ and paste that in the textfield there, enter the password you set for the extension when you created your MetaMask wallet. You should be able to see your seed phrase and any manually imported private key below if you click the Decrypt button.
Best Answer
Unless you find JSON formated text file with keys such as "id", "cipher", "ciphertext", "salt"...etc, I don't think you would be able to recover. If you find it, you will be able to extract it via not just metamask but also other tools, assuming you know the correct password. You need either that or your mnemonic. Password is just missing information to extract your keys from the file I mentioned. Dig deeper through that folders, hope you find it :)