I'm with homicidalmatriarch. It sounds like you have had a really hard week, but hopefully the next one will be easier. Can you just buy food that you slam in the oven for a few days (when you have your appetite back), ignore the washing/housework as far as possible, and get a few early nights in? Does your DD still sleep during the day? If so, can you BF the baby and have a doze at that time too? I have been there too, and it can be really hard at times. I don't think I used the vacuum cleaner for about 3 months earlier this year (not that I missed it ). Cut corners where you can in the food prep/housework areas, it's not going to matter for a few months.
Also, do you have any friends who can support you for a few months? Perhaps they could bring their LOs round to play with your DD whilst you slope off to bed for an hour?
Hope you are feeling more positive about it now. FWIW I did mixed feeding with DS1 from about 14wks, and successfully continued BFing till 11mths, so introducing one bottle a day once your milk is well established is not going to jeopardise your continued BF IMvHO.
However, if your milk supply is a bit low at the mo (because of your not eating in 36hrs and being under the weather yourself), I suppose it might be best to keep offering BFs and get it back up.
I know it's a hard call to make, I agonised over it for ages with DS1. (My main reason then was because I was returning to work quite early and wanted him to get used to taking some formula from a bottle). Have exclusively BF DS2, but I have a bigger age gap between the two, so was able to "persuade" DS1 to let mummy sleep on the sofa in the afternoon for 40mns whilst he watched cbeebies.
Take it easy, and don't be too hard on yourself - you cannot expect to be doing everything with DD that you did before you had your second LO. Not in these early months. Good luck to you.