It is reasonable to want to keep them for use for a lodger - you've already said that you're in a difficult financial situation right? Well... having a lodger would help you with paying rent. Explain to your mom your plan with the lodger - how much you expect to get per month in rent from a lodger.

Then explain that when you get caught up on any bills you might be behind, or purchase things which are truly necessarily (like say, getting a new pair of glasses because your prescription has been changed), then tell her that after then you will consider purchasing a dresser and/or desk. And that when you've been able to purchase those, that you'll gladly give the ones you have now to your sister.

I think it sounds perfectly reasonable to have a desk in a room for a lodger.

Ad like Perna said - contact your sister. Go right around you mom and explain your plans with your sister. As you're on benefits, clearly a lodger would be beneficial to you. And a lodger is highly unlikely to rent if they have to furnish the room themself.