Hi Molly, I am also a new mum to a 15 week old. She also sleeps in a cot in our room. When she was the same age as your DD she didn't usually go to sleep until we went up to bed, we used to do the last feed anywhere between 9pm and 10pm and then put her down afterwards. Although at that age we mostly rocked her to sleep.
We did a bedtime routine of pjs, sleeping bag, feed at first, then added in a bath first when she was about 9 or 10 weeks. At first we were doing it quite late, at 8:30pm to 9pm, and we just gradually brought it forward. We figured it was most important to do the same thing each night, even if it wasn't at exactly the same time. About 3 weeks ago she started signalling to us that she was tired even earlier, and now we start bedtime at about 6pm to 6:15, and she's in bed by 7. As soon as we take her upstairs to get her undressed for the bath, she relaxes as she knows soon it will be bedtime!
So I would say continue with your routine, but maybe don't worry about getting her to bed early. I know it's hard as you just want a couple of hours to yourself in the evening, but I think when she's a few weeks older she will be more likely to get tired early and settle herself.
Agree a dummy is worth trying, we did but DD just spat it out!
We also have the Angelcare sensor pad monitor, which means we're happy to have DD in bed in the evenings even though we're not in the room with her.
Good luck x