E.E.
I have a son with growth hormone deficiency (GHD) and we do the nightly shots. It can be traumatic to get a nightly shot, but at 11, she can probably handle it pretty well (our son was 4 when we started, and we'd just adopted his next younger brother - and that brother was scared and angry since he was too young to understand why his foster family had handed him over to us - and I was pregnant - it was a messy year for all of us, but for the son who started shots most of all)
Anyway....I suggest joining Magic List on Facebook or finding them on-line if you aren't on Facebook. Parents there tend to be overtly focused on height, but it's a good forum for parents of GHD kids who are new to treatment.
I don't think treatment is a good idea if she does not have any underlying medical conditions (or suspected conditions), but there are endos who will treat for height only (at 5'1", I find that offensive, but there are plenty of OTHER good reasons to treat her if she tests positive for GHD, because the development of muscle and bones rely on GH too).
She will need a stim test (a 4 hour test w/ blood draws every hour, if I recall correctly) and an MRI of her brain (to rule out issues that would contra-indicate treatment). If she does have GHD, since she hasn't started puberty yet, she will probably be able to benefit from 2-4 years on the shots.
Good luck and feel free to PM me if you have other questions you think I might be able to answer.
ETA: I looked back at your other questions and saw the one about your daughter being sensitive. My son with GHD is quite sensitive as well. I don't know if that is related or if that might get worse with the shots. I'd say it did with our son, but he has a host of other behavioral challenges, and was young enough that starting to grow meant he lost a lot of positive attention over being like an adorable performing baby - at 11, I'd think your daughter would like that change).