"the problem is I can’t look after the kids I go to work before they go to school and I get back well after so I have no way to do it" there is plenty else you could be doing.
"to break up this relationship without talking when 2 kids are involved" THREE there are THREE kids involved - seriously I'm surprised she didn't do this earlier!
"she will now be a single mother of 3 and I don’t want her to struggle" and yet she's doing it! I strongly suspect she has less work now not more.
"without my name and income they would not let her stay there anyway" well they can't evict her for being a single mum! And by the sounds of things she's switched on financially and has got herself organised so quite possibly she CAN cover the rent.
Frankly given that op's tend to write in such a way as to favour themselves and yet it's clear from what op themselves has said that they weren't pulling their weight at home, was overly critical of his ex, didn't listen when she did tell him what was pissing her off and had until now (now that shit got real!) not even considered how life was for her!
She's got 3 young kids, an entitled unsupportive partner, works part time and NEVER gets a full day off nor even support to do anything for herself of an evening if she's not working (if she even had the energy she must be knackered!) and yet apparently she was unreasonable to say "enough is enough"!!!
Op an oft used cry upon hearing of such setups on mn usually from your partners perspective (seriously HUNDREDS of threads like this) is "does he think it's the 1950's?"
When actually you're not even doing as much as a 1950's husband and father would.
Honestly, read those links I sent you, read some threads on here from posters in the same position as your partner, grow up and stop pestering her to take you back when fundamentally you don't want anything to change.
You want to carry on only weighing in minimally of an evening and on a Saturday, doing whatever you like on a Sunday and leaving everything else to her!
She is absolutely right not to put up with that and if she posted on here she would indeed have been told that unless you changed your behaviour and attitude drastically & permanently (no just for a few weeks/months to get her onside!) to get rid!
Genuine questions:
How much were you expected to do in your childhood home growing up? Did your mum do the bulk of household chores and responsibilities or was she a single mum? Have you lived on your own and thus been responsible for all chores, bills, admin etc? When your 2 dc were young babies did you do night wakings, early mornings, teething & colic duties, nappy changes, bathing etc? If the kids are ill what do you do? What do you do if she is unwell? Do you take any responsibility for the admin relating to the kids? To the house? (Think I can guess the majority of the true answers to these).
She DID tell you, you just weren't listening.