Now that your baby is 5 months old... is she perhaps just going through some developmental changes or growth spurt? This can really throw a hiccup into their sleeping patterns and their intake level increases and they need to feed more.
If you are wanting your baby to sleep through the night... well, each baby is different and their "ability" to self soothe varies too. Some babies will not sleep through the night until later... my first child did not until she was about 2 years old plus. My 23 month old still wakes sometimes once a night.
I know it's hard for you, because of your insomnia et al. At the same time, I think all Moms go through this because we have to wake up at all hours to nurse our babies and it's naturally harder to fall asleep after that. I've been doing that for the past 6 years. With some babies, crying it out just does not work. I know, it did not work with my first, but I tried it because we were desperate. Needless to say, I figured out other ways to get my children to sleep...from co-sleeping to pacifiers (my son) to soothing them and just giving them what "they" needed. Each baby is different.
Or, at about this age, my son really took to a stuffed cow...which he likes to sleep with and cuddle. It really helps him to sleep and he just adores it. We see nothing wrong with a baby having a "lovey." Although some do not encourage this.
Also, we have a Fisher Price "Ocean Wonders" crib toy in my son's crib... he LOVES to put on the music before he sleeps and watches it. This has become his routine as well. We also put on a room fan, on low, for some white noise.
We also have a routine, pre-bed/nap, and stick to it everyday.. Same time for naps/sleep, and same routine. THIS has really helped in assimilating my kids to sleep. I've done sleep routines without fail since my kids were babies... even at their ages now, they still nap and go to bed fine. I breastfed both.. .so I know how tiring it can be, getting up for on demand nursings, and getting them to sleep and back to sleep. But this is the method I worked out for myself.
For my son, in his case, he likes having a few baby-safe stuffed animals in his crib... it distracts him, helps him to self-soothe, and he will eventually then fall asleep on his own. (He couldn't care less about his crib-mobile on the other hand). So this is what has worked, in his case.
Another thought, although I am sure you tried this too: but does she have gas problems? Teething? My girl was similar to this too... and she had gas problems too... (the Hyland's Colic Tabs, is great. My girl did not have colic, but it helped her gas problems....it's homeopathic and has no contraindications).
BUT as we also learned... she was also "sensory sensitive" to sounds, texture and temperature. (she is still sensitive to certain sounds/noise even at 5 years old). Thus, she was fussy and hard to put to sleep. Crying-it-out NEVER worked with her. She had more will than both me & Hubby combined. She would NOT succumb to that. So with her, I had to lay next to her until she fell asleep. Then as she got older, I just gradually made my presence less and less, and gradually got out of the room for her to sleep. But this was over time.
Another thought, as this happened with my friend. Her baby girl was next to impossible to put to sleep, and always wanted to be carried and was always fussy or crying. She took her baby to the Doctor... he examined her head to foot. What they found was a tiny blond hair (barely perceptible to the eye) which was tightly tangled & wrapped around her baby's toe...it was causing pain and was getting into her skin as she grew. They removed it, and her baby was just so happy and her "fussy" demeanor stopped.
With other friends, their baby was not getting enough intake. Thus, baby was always hungry, fussy, did not sleep well or for very long, and was never quite satisfied enough. For one Parent she didn't have enough milk. For the other Parent, her baby just was not suckling effectively. After they got that solved, their babies slept better, and was less fussy.
I don't have any miracle answers for you... but I know it's not easy. You are obviously doing the best you can and trying things that may work. Sorry for rambling, just some ideas that popped in my head.
take care,
~Susan