First of all, this is totally normal. Many of them do this in the early days, and my middle one did this all day and night for ages. So please do not panic or worry! Bottlefeeding her formula will make no difference at all and actually might be more difficult for you, because what we are talking about here is the maturity of her liver rather than anything else (a GP chum explained it to me once but I have forgotten the exact science).
Secondly, even if you didn't feed her for even as long as five or six hours now and then, because you were exhausted, she would be pretty much fine physically if she was a full term baby, so don't be scared of going to sleep yourself in case she needs you. She would just cry a bit, that's all. Babies are tough as old boots.
Thirdly it would be a bit odd if you weren't a bit depressed considering you have had so little sleep. So don't fret about this either because it doesn't mean a thing, and it does not mean a fully fledged depression bout is around the corner.
You've got three choices at the moment, as I see it. The first option is to put the baby in a crib right by your bed and just sleepily reach across to get her when she cries, and feed her sitting up or lying down, but trying not to nod off yourself. This is the most officially sanctioned route. People call it 'co-sleeping' I believe.
The second option is to let her sleep in your bed with you, making sure she doesn't get too hot or smothered by anything. There is professional advice about how to do this as safely as possible, which you should check out first. This approach is what I tend to do. I just yank a boob out and let them get on with it themselves while I doze.
Finally your last option is to express some milk and get someone to give her a bottle of your milk while you catch up with sleep. My DH used to do a six hour stint on Friday nights this way so I got at least one night's proper sleep each week. This is also officially sanctioned.
The more you try to sleep when she sleeps, the less awful it will be for you, but a bit factor here is the sheer number of interruptions to your sleep and the disruption to your own sleep cycle, which is why you are feeling wobbly and shaky.
Everything will feel better in a few weeks, I promise you.