That seems reasonable, however we need to be sure that the mutation is actually in a gene that codes a protein.
If you can find the mutation in the sequence, you'll know whether its in an Open Reading Frame or not. So we know that the mutation is in a gene that encodes a protein.
How do you know that the enzyme that nylon eating bacterium used to break down nylon wasn't formerly used for something else?
Because bacteria did not intrinsically possess nylon-eating activity prior to the mutation. If they'd had the nylonase enzyme from the get-go, all related bacterial species would have immediately been able to metabolize nylon just as soon as there was some nylon to metabolize.
How do you know it was not a mutation to a gene controller?
Direct DNA sequencing.
That seems reasonable but it doesn't rule out gene switching.
No, it does rule it out, that's the point. If it's just a matter of switching on a gene that all bacteria have, then every bacteria will switch the gene on in response to the environmental need. That's the point of having a "switch." A good example is the lac operon - bacteria can produce the enzyme beta-galactosidase they need to metabolize lactose, but they won't do it if there's no lactose, and they won't do it if there's lactose but also glucose.
It's only when they're in an environment where lactose is the only sugar that they'll switch the gene for beta-galactosidase on and start to metabolize lactose. Every single bacterium with this gene switching mechanism does it in response to an environment with lactose but no glucose.
On the other hand, only a small number of bacteria gained the ability to metabolize nylon in response to the presence of nylon in the environment. That's what proves that it was a new ability gained by mutation and not the activation of an intrinsic gene.