If the initial tennacy agreement has ended, then you can issue a section 21 notice that asks for posseion of the property without having to give a reason. You have to give a clear 2 months notice, so saying she pays the rent on the 20th of each month, then you could give her notice up to the 19th, and she would have to leave by the 19th June. But if she doesn't leave, then you have to take her to court and ask for an eviction order, and then when she doesn't go, you go back to court for a balifs order - all of which can take about 6 months. Asking for a high court order is expensive, so the rent has to be high enough to justify that path; hence the suggestion to offer her enough for a deposit on her next property - or start at half of that, and let her work you up.
In telling us about the CCJ, then you're obviously going for a section 8 notice, so best to give her a section 21 notice at the same time. For future reference, the tanant would be 2 months in arrears the 2nd time she missed a rent payment, and not when she misses the 3rd one, and the section 8 notice could have been issued in January, as missing the rent is a valid reason for terminating the tenancy agreement prior to the 6 months protected tenancy ending.
But the caveat over all this, is whether you (or the managing agents) have followed the correct procedures. Was a deposit taken, and then protected, and was evidence of the protectio given within 2 weeks of receipt of the deposit. Was a legionaires assement made at the start of the tenance, and was an EPC given as well. If you've missed out on any of these parts, then you can't issue a section 21 notice, but I'm not sure how the section 8 is affected.
If the agents issued the tanancy agreement, but hasn't followed the necessary rules, then you may have a case against them.