A professor who agreed to write my letters was known for having little sense of time. I ended up emailing her daily prior to the my first deadline even though she never replies, and kept calling her secretary asking her to remind the professor. She sent all of the letters only a couple of hours before the deadline ended (although there is a significant time difference I didn't think I could expect her to do it after midnight). Before she sent it though I was refreshing the auto-updated page like mad and near losing it since I didn't think missing the deadline was acceptable. I was conjuring up any curse I had in my head but of course kept my emails ultimately polite.
Although she really nearly gave me a heart-attack, after everything was done I felt really thankful to her for taking the time to send the letters because I know how incredibly busy she is!
Point is, I kind of know how you feel. If that professor actually did miss my deadline I'd probably hate her too so it was good she didn't! Yes you should've made sure it was received well before the deadline, but it sounds like your prof's attitude was horrible so you can definitely feel angry. Never think about burning that bridge though because it isn't worth it!
Also, once you confirm her letters are received you should let her know and thank her politely. Hope things will work out for you!