Definitely keeping out of the landlord bashing here as I can definitely detect undeserved negativity.
Sadly, even if you use an agent, some are better than others, and some (most) are pretty useless when things go pear-shaped. They are also pretty rubbish at referencing, tend to use agencies and often tenants lie and/or even if they pass the tests there is no guarantee they will pay the rent, or pay it on time. Best to make sure of your tenants' previous record before hand - check out the references yourself in future, particularly from former landlords. Alternatively, it is possible to get rent and legal protection insurance (which you can take off your bottom line) provided they pass certain referencing criteria.
Always easy to be wise with hindsight, but there are some things you can do now. The two months' rent rule is pain, but there is a discretionary ground to seek repossession based upon persistent failure to pay rent on time. Make sure you keep records of when rent was due, when paid, and demands made etc. Going to court is always a pain, and will further seriously make dents in your budget, but you should keep this option under review.
In practical terms, I personally would get onto the rent arrears and late payments as soon as possible and do not allow these people to mess you around. If you do, they will go on doing so. Agents are often really rather rubbish at being proactive. Obviously, you need to stay on the right side of the law, but you should make sure the agent has set up a direct debit/SO from the tenants (should have done this before letting them in) and as soon as there are any arrears, chase. Leave them in no doubt that paying the rent should be their number one priority, and that you are not subsidising their lifestyle. Do not hesitate to send a letter before action if you need to, and follow through.
In some cases, it may help to speak to them personally. Sometimes tenants really do have problems, and it is in everyone's interest to work together (for example, if they suddenly lose their job, relationship breakdown etc.)
Unfortunately, it sounds like you may have got a professionally bad tenant here, or just one that thinks it's your job to fund their own irresponsible lifestyle. It's not. Get yourself some decent advice (and don't necessarily rely on the agent). Join a professional body, like the RLA, they have a mine of information. And if you need to get your tenant out, think about instructing somebody who knows what they are doing like Landlord Action or at least read up on it and speak to other professional landlords.
Chances are, if you make it known at the outset, you will not tolerate arrears and are not a soft touch, your tenants will play ball.
But don't take it for granted. And make sure you/the agent carry out regular inspections and check for other breaches (like unlawful sub-lettings).
Most tenants are fine, some are really good. But the bad ones can be a real pain in the proverbial.