Speaking from experience as a daughter, mother, & grandmother, this is NOT a trivial matter! Although she loved us dearly, my mom had nothing but criticism for us. I became anorexic [before our doctor had ever even heard of it in 1974], and at 51 I still feel dowdy and ugly and like I never quite measure up (even though I'm very intelligent, talented, sociable, hard-working, and average on looks).
She's a young lady, not a little child, so having her period, being in a new environment, and developing physically has had a big impact on her life. She NEEDS to feel as if you're proud of her, not disappointed in her at this point. She doesn't need 'pushed', but 'encouraged'.
It sounds like it's time to really listen to her heart and find out what SHE needs, wants and expects of/for/from the rest of her life. Find out from HER what all is going on in her head, heart and body. (Other) girls this age can be terribly 'clickish' and exclusive -- especially to a 'newbie' on the block, and an individual (13 yr old GIRL, in particular) who's at the butt end of this behavior is very sensitive and vulnerable. Maybe her uninvolvement with others is legitimate. Maybe you (or DAD) need(s) to do some active type of play WITH her.
Her whole future self-esteem rests on what happens RIGHT NOW, so try to be totally encouraging instead of demanding or critical in the least (although it sounds as if you may already have been somewhat).
Take her to a nice restaurant -- just mom and daughter -- and let her pour her heart out without offering ANY advice unless she asks for it. And if you feel too close to the matter, emotionally (gripey, advice-giving, etc), have another laid-back lady friend who she trusts to take her out and get to the heart of the matter.