Funny enough, I have that same situation with my two daughters - who are now 27 and 22.
My 27 year old only likes shooting a .22LR, and shoots quite well with 22 rifles, pistols and revolvers. At one point, she wanted to buy a Ruger SR22 and both I and her boyfriend (now her husband) talked her out of it, as we felt it was not something she should use for protection. Now I wish we had not discouraged her, as she has nothing, and has less interest now than she did 5 years ago in guns. She could have carried that little 22 in her purse or car for a few years, then we might have moved her up to something more reasonable.
My 22 year old on the other hand, who learned to shoot with the same Ruger Mark II as her sister, has her CCW and carries a Ruger LC380, which was about the best balance we could find for her year before last, with recoil she could manage while still maintaining accuracy. She could not manage anything smaller in 380ACP, and could not manage the smaller 9mm pistols. She likes shooting my XDM 5.25 in 9mm and is more accurate than I am with it! She hates her fiancé's .40 pistols.
My advice is you are trying to teach your 11 year old is to make it "cool" as suggested by Bridget, and also to train her with light recoiling 22LR pistols, revolvers or rifles, so that she can gain proficiency with a more comfortable weapon. Only move her to something bigger when she asks. That's the mistake my son-in-law and I both made with my older daughter - pushing her to shoot .38/.380acp/9mm/.40 when she was not comfortable with them.
I am thinking of picking up a Ruger Wrangler 22LR revolver to supplement my Ruger Mark II specifically to have something "fun" to shoot for my wife or the girls.
Maybe if you got a nice little 22LR pistol or revolver and gave it to the daughter as "hers" (to be locked up when not in use of course), that would pique her interest - if she could train on her own gun.