Having just gone through this with the trailer I'm retrofitting for my Nordic, I've a little bit o' info to share...
First up - Carlisle no longer has anything to do with Hydrastar. The company you want to deal with is below:
http://hydrastarmarine.com/
The company is well informed on everything Hydrastar, and was started by a former Carlisle employee as explained to me. They acquired all rights to the Hydrastar product from Carlisle. I'm not certain, but I think the head honcho is the original designer of the Hydrastar units originally sold by Carlisle. They have all necessary parts to fix just about any problem found.
When I contacted them, I had several calls and emails with Nick. He's most helpful, and can help you troubleshoot things in short order. Some of the basics mentioned above are correct, but he can give you an exact method of testing that will isolate exactly where the problem is. The yanking of the emergency plug - assuming a charged trailer backup battery - is a good start, but it doesn't do anything other than isolate the problem to the trailer or vehicle side of things.
Also, they have updated their products. The short version, as explained to me - the Hydrastar units were not originally designed for marine environments, as that was not Carlisle's targeted audience. They quickly caught on for boat trailers, and Carlisle was quite happy to supply them to trailer manufacturers - but the new versions have been updated to better hold up in a marine environment.
My Hydrastar was doing the same thing - clicking, no actuation. For my problem, two things were discovered. First up - my Hydrastar actuator was toast. It was installed incorrectly by Float-On, placing it inside the trailer frame rail in a location where it was easily submerged when launching/retrieving the Ranger 29 that was originally on it. When disassembled, the level of corrosion was extreme, and I opted to replace it with one of the newer units. I'll be mounting the actuator higher up to ensure it can't be submerged.
Second - my original brake controller is on the "approved" list. It's an original Prodigy, and when used with the HBA-CAM adaptor - it's supposed to work just fine with the Hydrastars. It doesn't, and there's nothing wrong with the controller - it works fine everywhere else. I swapped it out with a Prodigy P2, and all works as expected.
Good luck - but start by contacting the folks above. I'm sure they'll get things resolved for you in short order.