Even if the OP could manage to farehop most of the way to work it wouldn't be worth it - they do check on those lines sometimes, even on the trains themselves, and it's not worth the risk of turning up to work very late because you broke the law.
IME the bus routes on that journey from North East to Central London can take an extremely long time and it's hard to predict how long they will take; they're not good for work with a definite start time.
So I'd suggest getting the overland train instead down to somewhere like Bethnal Green overground (not sure where your line goes to but if you start in zone 2 and stay in zone 2 for the whole journey then you pay the same price even if you change), or somewhere else that's still in zone 2, and walking from there.
It'd be about half an hour in busy traffic to walk from Bethnal Green overground to the city - busy traffic also slows down pedestrians due to more people on the pavement and longer waits at crossings. It would still be cheaper than multiple bus journeys and far quicker and more reliable. Put something good on your headphones and think of it as exercise that, unlike cycling a long way, doesn't require a shower at the end.
Going back, you could walk further to get to one slow bus that takes you all the way home for one fare - you have more flexibility there.
BTW, WRT JSA, if the OP didn't have enough contributions paid in for contributions-based JSA, either due to previous unemployment or previous low pay, then she would have had to claim income-based JSA and wouldn't have been entitled to it. Don't know if that's your situation, FiscallyChallenged, but someone else said that "something doesn't add up" based on what they thought they knew about JSA.
It could also be that you didn't know that you could claim, but that doesn't help you now. The jobcentre doesn't offer help to get to a new job any more anyway and crisis/community care grants are pretty much gone.