Will you tell me to rebuild it?+
Only if the evidence says so, and the evidence is written down where you or any other engineer can check it. This is the judgement you have most reason to distrust, because a rebuild is the largest engagement I could sell you. So the rule is published on this page rather than applied privately: a rewrite is only defensible when you understand the requirements better than you understand the code. The verdict is reasoned rather than asserted, so the argument can be attacked on its merits instead of taken on trust.
What if I do not have access to my own code?+
Then that is the first problem, ahead of the code itself, and it is why the checklist on this page comes before anything I sell. Secure everything you already have the right to access, and take legal advice before pressing for anything you do not. On the IP question specifically: absent an explicit written assignment, paying in full may only have bought you a licence rather than the copyright, and that varies by jurisdiction, so have your own lawyer read the contract. Where there is no repository at all, a diagnosis can still run against the running system, the deploy record and the contract. It is a weaker read and I will tell you it is weaker rather than dress it up.
Do I have to fire my current developers?+
No, and it is the wrong week to decide. The people who built it hold information that exists nowhere else, and they usually expect to be blamed for it, which is the fastest way to lose that information permanently. Establish the facts first. If a change of team is the right call it will still be the right call in week two, and by then you will be able to explain why in terms the team, the board and the next vendor can all follow. Working alongside the incumbent is often the cheaper path; what it needs is written boundaries before the first change, on merge rights, releases and escalation.
How long before I know whether this is salvageable?+
Five working days for the written verdict. A production system that is actively failing gets a first read sooner than that, because stabilising something that is losing money now does not wait for a document. The reason it is time-boxed at all is that your money is burning while you decide, and an open-ended audit bills for the deciding.