LHL - don't you DARE feel awful!!
It's not you (everyone), it's our stupid modern Western culture expecting one person (whether in a two parent or lone parent situation) to look after a child 24/7 in isolation. It ain't natural. Don't beat yourselves up. They say it takes a whole village to rear a child - and doesn't it ever!
This country in particular is very unsupportive of families, especially lone parents, and it's appalling the way we as a society leave lasses like you struggling like this. We should be ashamed of ourselves.
Never mind NHS Direct and Childline - we should have 'Dial-a-Gran', 'Emergencyuncle.com', 'I'm-Gonna-Kill-the-Kids-Send-Someone-Round-NOW' and 'Mums-(Climbing-the-Walls)-Line-HELP!'
Short of that, sod the cleaning for a start and I hope none of you EVER do any ironing. Spend the time on the two year old, they're very demanding at that age - they just are - use distraction rather than confrontation, it makes your life a whole lot sweeter and keeps them sweeter too. The less you have to use the 'N' word, ('no'), the fewer screaming fits you'll have (from the toddler, not you) so try to engineer-out situations where you have to say 'no'. This is NOT 'spoiling' them, it's sheer common sense - they're two.
Make your home as kid-proof as possible and keep stuff they shouldn't touch out of their way, (get a stairgate, Sparkle and electrify the door handles).
Don't take them out when they're tired or due a nap unless they can sleep in the pushchair.
Play nice soft, calming music (Dvorak's 'New World' rather than Metallica' - never mind if you're not into Classical, it's calming) and don't feel guilty about parking them in front of their fave dvd often occasionally, though personally, David Attenborough rather than Brattz (one of mine is now a Zoologist with a Class 1 from Bristol uni, thanks to Dave). And while they're occupied, put your feet up and have a nice cup o' tea (or a glass of merlot), watch the dvd with them, DO NOT do the hoovering.
If you must do the cleaning, give them a little (harmless) job to do too. They'll make a hash of it but that's not the point. It makes them feel they're helping and sets a habit for life. This is a mistake I made (i.e. not doing it) but then I had six on my own and it was easier to do it myself than oversee that many at a time - and I had a farm to run and none of them were any good with a chain saw.
Do you have a paid job? (Yeah, yeah, I know - it's enough of a job with a toddler) but, if you can weave your way around the shitty benefit system and get some part-time work (that suits you and that you like), it'll get you out of the house and if the childcare's great it might be what your litt'lun needs too for a few hours a week. If you're on Income Support, they may take your wages off pound for pound though I think you can do so many hours without it affecting benefits much - watch how it effects the Housing Benefit and Council Tax though - I'm assuming a lot here, so forgive me if I've got it all wrong and I just sound patronising. The point is it sounds as if you need a break and while you may not be that much better off in a paid-for job (as opposed to the unpaid job you're doing already), it's a different environment and it'll be a holiday compared and your litt'lun will be (hopefully) happily worn out from playing with other tantrummy two year olds and someone else will have had to cope with it all - reeezult.
Keep your mates. If you can't always organise it so's you have a night out once a year week (it's expensive!) ask a few of them round to yours for pasta and alcohol - bring your own. If you can wangle it that dc is at dad's/grans - and so are theirs - great - go out. If not, ask them anyway, with their kids. Get some blow-up beds and travel cots - (or some tents if you have a garden - the weather's lovely - you can all sit round the barbie/fire - kids love that - you'll be amazed how quiet they can be staring into flames, as they drift off to sleep cuddled up in blankets, just listening to the grown-ups chatting)- put the babes to bed - have a laugh about your ex's - get pissed.
That last suggestion is recommended.
Failing all that, have a few more - they tend to keep each other occupied (just kidding).
The first one is always the hardest, simply because it's nothing like anything you've ever done before. You've had no training and no warning. You get no pay, no pension, no appreciation, no help and no thanks and everyone thinks (including you) that because you gave birth to it you should know automatically how to manage and if you don't you're a failure. What we all fail to appreciate is that nature never intended anyone to do it alone. What this society fails to appreciate, or continues to ignore, is that two parents is not enough so stop giving lone parents a hard time and cut them some slack. We're doing a damn fine job rearing the next generation, a hell of a social contribution that no one else would do without a big fat salary - and that deserves at the very least support and respect.