Ghostie - I really sympathise with your sleep deprivation. It is torture, isn't it? However, I'm afraid 5am is a normal time for a little baby to wake up. Both of mine would wake at this age - it was no use leaving them because they weren't tired anymore. Also, if she settles in your bed, I would just bring her in with you, or maybe have her in a cot beside you (I know this is the current advice). If she is in bed with you, be really careful to make sure she can't go under the duvet or overheat and that neither of you has been drinking too much (!).
I have had two boys. It was utter hell with my first because he just wouldn't sleep. I kept him in bed with us for the first four or five months, then moved him to a cot and slept beside him (I was worried about cot death and had heard that it was good for a baby to have you with them). The sleep remained an issue and I still have my 'sleep diary', where I noted down his waking times so I could show the health visitor what he was doing. My sleepy scrawl shows how exhausted I was. Sometimes he was waking up as soon as I put him down and I was beyond looking for a reason anymore - I just wanted him to let me sleep.
I really don't think there is a lot you can do when they are 5 months. What you need to do is get sleep during the day when you can. My son (who was breastfed) continued to want a nightly feed for the first year at least. In the end, I used to breastfeed my son to sleep at midday and he learned to take a long nap then, which was when I would have my own nap. This enabled me to survive the disrupted nights. Teething is a real problem and caused lots of waking and crying. Calpol was often the answer to that.
The same thing happened with DS2, but I was better able to cope this time because I was expecting it. It was still hell and I often couldn't catch up on sleep during the day because I had another child, but the difference was that I knew this phase would end. I also used Medised when he was ill and blocked up with a cold, which I'm afraid they now won't prescribe for children under six because people were misusing it to get their babies to sleep. I was careful to only use it when he needed it, but it had the wonderful side effect that he slept well and I was guaranteed a few hours' sleep myself.
A friend of mine with a four-month-old baby feeds her son at about 5, then hands him over to her lovely, helpful husband and goes back to bed for two hours before getting up for the school run. I have to say my husband wasn't up for this at all, so I just struggled on, but I think this is the answer. You need to fit in with your baby and grab sleep where you can, rather than expect your baby to fit in with you.
Five months is a really tough time, I think. You are through the initial euphoria of having your wonderful new baby and the hard work is wearing you down. It seems relentless at the time. But it really will get better. Just hang on in there.
One last thing - I did have cranial osteopathy on my second son, which seemed to help a bit. An osteopath would say that babies sometimes suffer pain from birth trauma (not necessarily obvious trauma to us) and that they can realign the bones so that there are no aches and pains bothering them. It's worth a try.