Is there any sin on my mother for spending a sum of 800 Egyptian pounds that she found several years ago, and what should she do if there is sin upon her?

The mother was obligated to identify the amount she found for a period of one year. If, after a long duration, she became overwhelmingly certain that finding its owner was hopeless, then identification is no longer obligatory. However, she must repent to Allah the Almighty, and she must donate this amount as charity, while guaranteeing it to its owner should he appear.