There are really only two ways of dealing with this. You either pay them or if they do issue legal proceedings, you defend them.
You have a defence that the landlord has failed to mitigate his loss by firstly failing to allow you to find another tenant, secondly by him failing to try to get another tenant and thirdly, failing to do the checkout inventory.
The signature in all honesty is neither here and there except that it commits you on paper whereas the commitment arises in any event under the terms of the tenancy agreement.
The agent cannot instigate the legal proceedings on behalf of the landlord because they are not solicitors. If the threatening letters from the agent don’t get money out of you, then it will have to be the landlord that takes you to court although the agent can actually issue the proceedings in the landlord’s name, however this gets to court, the landlord is going to have to do attend court. If the landlord of the property is down as being the name of the agent, they can bring proceedings because they are then not acting on behalf of the landlord but on behalf of themselves. It would be unlikely for the landlord to be the agent (although the landlord can use the agents address) and if the proceedings end up in court with the agent representing the landlord, you can ask for the proceedings to be thrown out on the basis that the agent cannot act for the landlord as they are not solicitors. They would then have to reissue proceedings so all it will do is buy you time.
You do have a reasonable defence however that the landlord is failing to mitigate his loss for the reasons I mentioned earlier so it would be worthwhile writing to the agent in the same terms.
Can I clarify anything else for you?