Go to speak to a lawyer that specializes in intellectual property. They can at least advise you on 1. how to approach the subject of royalties, and or terms and how much to ask for! and 2. whether you have strong case proving that you own the rights.
Disclaimer: My post is not legal advice. I am not an attorney .
I would be inclined to say that YOU own the design. Legally there may not be much you can do if they decide to screw you over, but I would say that it is worth attempting to protect your interest in this situation.
If you create a work as an employee, your employer generally owns the rights.
However, if you contract a photographer for example, they traditionally retain the rights to the product of the photo shoot, unless it is specified that the rights will be assigned to someone else in the contract.
I can't offer much about information about design patents, because that is a can of worms I am not familiar with.
Technically copyright law affords some protection to sculpture, architecture, and there are even specific mentions to furniture, especially unique elements such as carvings, or turnings. Not so much for basic forms that are common. People are often not aware of this, but you can learn all sorts of interesting things if you actually read a law.
Also copyright is automatic (due upon creation)That however does not guarantee that you would be granted a registration which would likely be required to seek damages.
This protection is not nearly as explicit and routine as such for print works, music, photo/video, and other visual mediums. Basically this is getting into the realm, where you would need advice from a lawyer, on what may or may not apply in your situation, and whether you are in a position to do anything.
Good luck! I hope you make some money from this.