Getting a DS Lite is the best solution I can think of - most are affected by broken hinges and all it takes is getting a nice case for them. In addition, the DS Lite has a GBA slot while DSi Mode doesn't give a lot of benefits unless you have DSiHax installed. You'll get the most bang for the buck if you get a DS Lite, a DSTwo to go with it and optionally a SLOT-2 cart for GBA support and/or extra RAM.
By the way... why do you specifically want a broken one? They're not really that expensive these days anymore, y'know.
By the way... why do you specifically want a broken one? They're not really that expensive these days anymore, y'know.