I think that there are (or maybe should be) two separate things going on here.
One is the relationship that you would like to have with your SIL's family. Telling them that you are sad and lonely at Christmas could be perceived as manipulative and passive aggressive. Your texts after you sent the money were both acknowledged with thanks, but you are not happy with the replies, as you wanted either an invitation or a conversation with your niece. Neither of these things are unreasonable in themselves, but probably shouldn't be expected as payback for a gift, which should really be given without 'strings', even when it is as generous as yours was.
The second thing is that your niece did not thank you for the money. At ten, she is probably too young to do this entirely on her own, so the blame lies with her parents (and not just your SIL - what about your brother's role in this?). At the same time, it is just over a week after Christmas, and it could be that your niece is writing thank-you cards to everyone who sent her gifts. These will take time for a ten year old to write, and there have been Bank Holidays with no postal service in between Christmas and now. Also, it could be that your SIL feels that you have been thanked, as she replied to two texts about the money, and she may be doing the 'thank-yous' on behalf of your niece, who is still very young.
I don't think that there is anything wrong with asking for contact details (email, WhatsApp or Messenger are all possibilities) for your niece so that you can 'chat' to her direct, though. Now that she has an iPad, she will probably love to have a 'penpal' to share photos and little messages with, and it will take some of the onus off her mother, who may well be busy with all the day to day things involved in working and bringing up a family.
That way, you could become more of a part of the family, and it could lead to invitations that come more naturally than before.