Your tool for gettting her to sleep works, but it is not teaching her to fall asleep herself..thus when she rouses in the night she wakes fully as the "going to sleep" sensation: singing and stroking, is no longer there and she needs you to "put her to sleep".
Any method of sleep training will work eventually: controlled crying is not to everyone's taste but is often the fastest (between 3 and 5 nights) if done consistently. Controlled comforting is similar but you resettle the child with a cuddle each time, takes self control to back out and keep to the plan to gradually reduce contact though and averages 7 or so nights and does not guarantee no crying at all. Other methods of gradual with drawal take variable amounts of time.
Your problem is that you are so exhausted that the mere idea of sleep training is probably making you feel weepy. And also you love your daughter and probably over identify with her (probably imagined) sense of abandonment.
being able to put go to sleep yourself is a vital life tool. this is something that your daughter needs to learn. You are doing a good job as a mum and this is the next skill you are going to teach her, just like you helped her to use a spoon, cup, climb stairs etc.
When you decide the time is right gird your loins and get a friend on standby on the end of a phone, pick a method that suits you (either the shorter sharper burst of controlled crying or one of the other methods) and go for it.
You will be successful, and once you have helped her to gain this skill you will be so proud of both of you...not to mention rested!
Good luck