Hi and welcome to the Forum.
The first thing you need to consider is that your partner needs to legitimise his stay here; if he's illegal and gets caught, he'll be deported with no right to return for 10 years, which is probably not the desired outcome. Whilst I agree with the advice you've been given above, that will cost you money and there is possibly another legal route for you to consider before you start spending cash. So what to do; my advice would be to go to the Citizens Advice Bureau, they are experienced in dealing with such cases, and are confidential and free to use; this link will take you to the relevant part of their website that deals with this.
The UK Government also have a website on the subject, I wouldn't advise you to contact them unless your partner wishes to leave voluntarily, but here is a link to their website.
I hope this helps.
Cynic
Expat Team