This isn't really something we would say. We don't refer to most clothing items as 'broken'. Maybe, "The slippers have a hole in them." or "The slippers are worn out." Even then, we don't send shoes for mending, we just buy new shoes or slippers.

I'd agree with Johnny that you'd be unlikely to say that the slippers were broken - you would say more precisely was wrong with them or what needed mending. Perhaps there's a hole, or that part of the slipper was worn out, or missing, or had fallen off, or needed to be replaced. In the UK, at least, we do still have footwear mended if it's cost-effective. Slippers probably aren't worth mending, but it's possible for this to be done. For example, if the soles of your favourite slippers have worn out you could have them replaced. And in this case you'd use the causative form of the verb - have or get + object = participle eg 'Get/have them mended'.