Many people do their own speaker repairs and it is many times perfectly satisfactory. The folks who do a lot of it are musicians who seem to bang their equipment around quite frequently.
The problem with your JL sounds like the dust cover and not the cone itself, so you are lucky in that respect. Repairing the cone is more involved and it's rather tricky to duplicate the thickness, rigidity, and original elasticity of the cone. It's usually best to simply replace the cone, a process called "reconing". But gluing the dust cover is perfectly acceptable and not much of a job at all. Don't try super glue. It won't work. I'm not sure about rubber cement, though, but I've never heard of anyone using it. Many pros use silicone sealant. Either black or clear looks best for speaker repair. A somewhat better product is a tennis shoe repair product called "Shoe Goo". It's a little lighter and remains more pliable than silicone sealant, making it a better choice for lighter speakers. You can usually find "Shoe Goo" at larger sporting goods stores. If you want to use something that is made especially for gluing speakers, here's a link to a product called "Prosound Loudspeaker Recone Adhesive" and is advertised as the same black adhesive as that used by many major speaker manufacturers. It's $19.99 for 8 fl oz, more expensive than silicone sealant. Is it better? I don't know. But if it were MY JL, I'd probably spring for it.
You may want to gently remove the dust cover entirely from the speaker and make sure that there are not any tears in the cone itself. If there are, you should repair them first.
Also, when you reglue the dust cover, wait a MINIMUM of 24 hours for the glue/sealant to set up and cure properly. If you start the sub thumping before the glue has fully cured, you'll have the same problem again. One other thing, use just enough glue/sealant to do the job. Don't think that more is better. A consistent thin layer on the entire surfaces being joined is best. A gob here and there will not hold for very long.
The alternative is to send it out for repair. And since it is a JL, I would not send it anywhere but JL. Be advised that is not going to be cheap. And it's going to be time consuming. Probably at least 25-50% of the cost of a new sub and probably a minimum of 3-4 weeks, sometimes longer. And don't forget, you'll pay shipping both ways.
Hope this helps.