Is the questioner obligated to repay the money her brother spent on her, after he demanded it 13 years after their mother's death, and is it permissible for her to perform Hajj under these circumstances?

Your two brothers are not entitled to demand from you what they spent on you from your parents' money. As for what your elder brother spent from his own money:

1. If your father was in financial difficulty and unable to provide, and your brother was well-off, then providing for you was obligatory upon him, and he is not entitled to demand anything from you. 2. If your father was well-off, then providing for you was not obligatory upon your brother. If he spent on you with the intention of maintaining kinship ties, he is not entitled to demand repayment. If there is a dispute between the spender and the one on whom the money was spent regarding the intention, then the word of the spender is accepted with his oath, unless there are indications that the spending was an act of kinship. In the event that he is entitled to claim reimbursement for the expenditure, he should claim it from your father, not from you.

It is advised to reconcile differences, maintain kinship ties, and hasten to perform Hajj if you are able.