It is too short for a long term indefiinate contract on an apartment.

 Granadainfo has some places with good prices for long stays.

A good combination could be the Fatima Apartment at
840 euros for 4 weeks (30 euros per night)
Info here:
http://granadainfo.com/fat/

There is a Spanish school just around the corner at
Carmen de las Cuevas.
Info here:
http://carmencuevas.com/
They also do Flamenco dance and Guitar lessons.

Another cheapish option is Richard's apartment
896 euros for 4 weeks (32 euros per night)
http://granadainfo.com/piso/

It is always a good idea to ask the owners of apartments to give you a discount for long stays at low season.