Okay. First of all, please stop beating yourself up. You're not a shit mum. You obviously care a lot but are worn down, which makes everything harder. When you have a sensitive or temperamental child, it usually takes a lot of refinement to find a good routine, and even then many things can send it out the window.
You also sound anxious (obviously) and perfectionistic, as well as highly sensitive yourself. I don't say that as a criticism but simply something you need to be able to acknowledge and accommodate for yourself. I don't think there is any known way to make a sensitive person less sensitive. I'm going to take a guess that at work you're highly capable and well-thought-of and don't feel stressed and/or bored by work the same way you do parenting a toddler, and this makes you feel bad about yourself?
Society often seems to prescribe that in order to be considered 'good', mothers need to want to be around their children 24/7, never lose their patience, never feel overwhelmed, never want to have time for themselves. While some women might feel this way, this is an unrealistic expectation for a lot of women.
Toddlerhood can be an exhausting time as you often don't get a real break during the day once they drop their nap, although it can also be freeing in that you no longer have to accommodate the nap drama on a daily basis. Swings and roundabouts.
But the challenges/freedom of no-nap time is partly why people start sending their kids to preschool for a session or two a week! You seem to be getting decent sleeps and a bit of time to yourself in the evening... if you find you can't make it through a full day of parenting without starting to feel frazzled and annoyed at your child, maybe you need to look at a morning at preschool a couple of times a week, or getting a student/nanny to come in for a couple of hours in the afternoon a couple of times a week so you can go out for a walk, or read a book or go for a swim or even get a part-time or volunteer job – whatever is going to make you feel more 'you' again and capable of handling the tougher days with more grace.
On a practical front, you seem to be having a few quite decent days among the bad ones and I actually think your sleep routine is pretty solid on the whole. My second was a terrible sleeper – I would have considered 5m of crying before going off to sleep by herself followed by a 1.5hr nap a near-miracle. Fortunately as my first was a better sleeper, I knew that it wasn't me 'handling things wrong' but just a different child.
I would try cutting or dropping your daughter's nap. I know she's still young but mine was starting to drop her nap by then. She had her last daytime nap the day before her second birthday and her night sleep actually improved a lot after that.
In all honesty, 45m of crying does seem like too long to leave a young child. I think once she goes past about 5-10m, you either have to accept that it's going to be a shush-to-sleep day, or just pick her up, skip the nap and try putting her down an hour earlier for bedtime. If you desperately need some down time, I don't think there would be anything wrong with putting on the TV for an hour or so and just chilling out with her.
Something I found helpful to repeat to myself when I was feeling frustrated and worn out by managing my daughter's sleep was "She's not giving me a hard time, she's having a hard time." It reminded me we were on the same team.
Good luck. Toddlerhood is just often hard and sometimes all you need to know is that you're not alone and many perfectly capable mothers struggle with it. I personally found the baby and toddler years really hard and parenting became far more rewarding once they got to school age. Now they are 7 and 8 and I find them absolutely delightful.