My guess is the chain separated from the arm that is supposed to pull it up when you hold the handle down. - This can often be reconnected with somethng such as a paper clip if it is the actual problem. There is not too much in the tank taht can go wrong - so worst case scenario - you would probably pay an hours worth of work to a plumber. May want to consider taking a few pictures of what looks to be broken - bring it into nearest Home Depot/ Lowes - or equivalent home improvement store and show it to an employee as to if they can make a recommendation.
In regards to nice traits I look for - it would take me a while to think about that part - and I know the above fixit advice you are looking for soonish :) So I will end here.