Just some thoughts:
1) its her age stage & development.
2) yes, not all kids are that way...
3) she seems to have no 'interests.' Meaning, she has no idea, within herself... on what to do. Are her activities things that SHE is interested in, or things that you signed her up for????
4) It is good being 'bored.' It makes the brain work and allows a child to get in touch with their feelings... even when doing NOTHING at all... it can help a kid.
5) She has to be taught... HOW TO get along and be a "sibling." Do you all teach her about how to get along? She seems to perhaps.. NOT have adjusted to having a little brother. Do they even play together? I have 2 kids... a 3 year old boy and a 7 year old girl... they are 2 peas in a pod.... they LOVE playing together, get along, love each other and really 'understand' that they are siblings... and we are FAMILY.
6) Your girl seems to be lonely.... you said, she seems to get "tired of us spending so much of our energy on him...." (her brother).
7) There are 2 types of people: "internal locus of control" and "external locus of control" type people. Look it up online.... your daughter 'seems' to be the type, that goes according to EXTERNAL based expectations and reasons. But a person can be taught... to be more self-reliant and fulfilled.
8) she's a kid... their every waking hour does not have to be an orchestrated symphony of "things" for them or for them to do. I would.. have her 'study'... read/do projects/write in a journal/do artwork... ANYTHING that will keep her school skills practiced.
9) She can also learn to cook. Over the summers, my Mom would teach me how to cook. Or teach me how to sew. I LOVED it.
10) ASK her... "why" she is not happy.... maybe it is her age, maybe it is because she can't do anything on her own, maybe she can't think of anything on her own, maybe she doesn't even know herself... maybe she needs more structure... not just "activities....."
11) Maybe, she needs to have a relationship with you/her parents etc. She just seems to do stuff or not do stuff... and entertainment or be entertained. But, does anyone really know her or understand her or have a relationship "with" her? Most times, kids are just told what to do/scolded/punished/given orders all day. But the parent does not really have a 2-way RECIPROCAL relationship or communication with the child. For me, when my 7 year old girl is fussier than typical... I sit down with her... we talk stories, I listen to her, I ALLOW her to express herself without my judging her... I laugh and giggle 'with' her... I ask her what she thinks... and I spend time with her directly.. not just having her "do" things or "go places" or join things. Not 'external' type things... but I spend time on HER.... and she feels "bonded" to me that way and we learn about each other. We are really close. THIS alone... affects my daughter positively... and makes her more grounded and even-keeled. We have a 'relationship'... not just a parent/kid do this-do that rapport.
12) Your girl seems typical. She is 7. She doesn't know what to do with herself. But.. you said that your 2 year old takes so much energy. Well... most kids do. And boys are active. I know, I have a 3 year old boy. BUT... I have always emphasized "family" and their relationship... and they thus get along well and with each other. We do not just focus on my son... just because he is younger. An eldest child... needs a TON of 'attention' too... even more so.
13) I would create a routine/schedule for her, daily. Have her practice school work too. Do chores and allow her fun stuff too. But have a routine daily. Or have her friends over and have play-dates for her.
Not all kids, can entertain themselves. But they can, if they try.
Just some quick off the cuff thoughts.
All the best,
Susan