How can I help her to go to sleep on her own?

DD1 (27 months) has always been a poor sleeper. She sleeps in our bed and will only go to sleep if DH or I lie down with her. This is okay at night since DH goes to bed the same time due to his early starts. But if DD1 wakes at 4am when DH gets up and I'm up feeding DD2, then she won't go back to bed until I do. Feeding DD2 while lying down isn't an option as DD2 gets too much wind. Also at naptime, if DD2 isn't asleep it makes getting DD1 down for her nap impossible. I can be sitting in the same room rocking or feeding DD2 but it's not enough for DD1. She will only lie down and go to sleep if I lie down with her. The lack of regular sleep times is really taking it's toll on her (and me) but I don't know what else to do.

She's so stubborn and has to have everything exactly the way she wants it and no amount of gentle coaxing (or frustrated grumbling or yelling) will change her mind.

I don't really have any advice, my 5 year old still needs help going to sleep, but our girls (my DD2) are nearly exactly the same age (mine is 2 days older) and she gets DH to pat her to go to sleep.

I don't pat her though, I sit in there on her bed, and be firm (but not angry) that I'm not going to pat her to sleep, but I will stay in the room until she falls asleep. If she gets upset, I just remain calm, and keep reassuring her that I am helping her to go to sleep. So you could try something like that - I find that I have to rock DD3 to sleep during the day while sitting on the bed trying to get DD2 to sleep, and if she wasn't used to me not patting her, then I would have serious issues too.

But if you ever find out the secret formula to getting them to fall asleep on their own - let me know! LOL

If it was me, I would get tough. I would put her in her bed and leave the room, if she gets up, you go and put her back down. They say it only takes 3-5 nights to change a pattern... so I'd be happy to go through 3-5 nights of hard times. She'll only be stubborn as long as she thinks it'll help her to get her own way, or that she will always get her own way. Once you show her that you're in charge she'll start to respect what you say more.

We did this at a very young age and have never had any issues with sleep. He goes to bed at 6:30pm every night and we don't hear a peep out of him until 8am.

Good Luck!

Think I'd probably try the tough love approach too mainly because of her age. My middle child (DS2) was a shocking sleeper then at 2.5 urs old he all of a sudden would sleep. Think I just got a bit tougher on him with being pregnant and not wanting to be dealing with him waking and feeding a newborn.

That sounds hard, especially having two kids to think of.
My son could not be left alone at sleep time till he was about 2, and from then we found he gradually got the point where he was happy to go to sleep by himself (following a rather intensive bedtime routine, I might add). By 3 he was telling his daddy to go away when he was ready to sleep :)
We did nothing to push this and I think the main thing that helped was time. It really is normal for young children to need help to sleep. At this age, though, you may be able to tweak things and perhaps gently encourage more independent sleep, if she's ready.
What happens if you leave her alone briefly? Can you negotiate sitting by the bed, rather than lying with her? Those first little steps are often the most difficult - do you have someone to help with DD2 to give it a go for a few nights? You may find that if you can get her to sleep sitting with her, rather than lying down (which might take a while), that the next step - just ducking out quickly to go to the loo or whatever - will be easier, and from there it's not such a stretch to her going off to sleep on her own.
Has she finished teething?

how are things now?

How about increasing the distance between the two of you (when you lay down) so that she is less reliant on your touch to go to sleep? Then work on sitting rather than laying.