It's not that they've outright refused to pay, it's just that they're
a) past the payment due date specified in the contract (which was 21 days from receipt of invoice ... really lenient)
b) not responding to emails as of about 2 weeks ago
Talk to your lawyer. Don't waste time faffing around with this. You don't have to sue them but you do need to get advice from a legal professional who specializes in this kind of thing.