Hi, you shouldn't need to stay with them. Once they have settled you can walk out, if they start to cry again you will need to go back in and start again. You can also do a combination of pu/pd pat/shush, until they have got the hang of it. No you won't have to do it forever, if you stick to it and are consistent, the time it takes doing pu/pd will decrease.
If you think she is teething and in pain always make sure she has had pain killers and give her the comfort she needs. You wouldn't do pu/pd if she was unwell.
If she has got used to you being in the room you may need to gradually move your chair further away each night until you are out of the door, or you could just do pu/pd.
Although I have made reference to pu/pd, at 13 months, it is better just to do the put down bit. They can often settle quicker out of your arms then in and they are getting heavy!