I can try and provide some advice on the law, but I cannot claim it is 'legal advice' since that would imply a duty of care towards you that is not appropriate for an internet forum.
Actually registering a design is fairly straightforward and you can find out how to do it on the UK Intellectual Property Office website
www.ipo.gov.uk. However you will have to pay a fee based the number of designs involved. Fees start at £60.
However, from what you have said I don't think your registration would succeed, or at least it could subsequently be challenged by the owner of the company. This is because design right, like copyright, is not always automatically given to the creator if he produces the design in the course of his employment. Here's what the IPO site says:
"If you are the creator of a design, you will be regarded as the owner of that design and entitled to apply for design registration unless:
* You have been commissioned for money or moneys worth, in which case the commissioning person is the owner;
* You created the design as an employee in the normal course of your employment in which case the employer is the owner"
The fact you were on an internship clearly affects your employment status at the time - and I am not sure of the legal position on that aspect - but you were 'commissioned' to design the furniture for money, although I assume you were not actually paid. In other words this is more about your legal employment status and the contractual relationship which arose when the employer asked you to design the item. You could try contacting the IPO for advice, but really I suspect that you would need to speak to a solicitor about this, for the reasons I have mentioned.
In the end this is going to be expensive for you and the employer if you choose to litigate on a matter of principle. I suggest that you contact him and try and negotiate a payment for your work and in return you agree not to take the matter of breach of contract further. Obviously you would not then be able to claim the design right, but then you wouldn't have had that right if you had stayed with the company and been paid as originally agreed.