Seems to me all 3 of you are at fault -
You for agreeing to something without realising what you could be letting yourself into;
The landlord for not getting you to sign a proper guarantor agreement & for not dealing with the mouse problem;
Your friend for moving out & dropping you in it by stopping paying the rent.
The 3 of you need to get together & sort it out. The landlord needs to deal with the infestation, & your friend needs to abide by his contractual obligation (i.e. to pay rent until his tenancy expires or the landlord finds another tenant) in return for that being done.
If none of that is possible the landlord may try to get the money from you. He will have to argue that you entered into a verbal contract, but if you were given no details (such as how much rent you were guaranteeing and for how long) he may have a problem getting this accepted if it goes to Court. nonetheless, a verbal contract can be enforced - it's just that it's more difficult to do as it depends on your word against his unless there are witnesses.