I usually use Beninter to book my holiday apartments, because they are more flexible about changeover days, but for a long term let Gestaltur would probably be a better choice as they are cheaper. If you let them "allocate on arrival" they charge 527euros per month (£360) for stays of three months and over.
Phone +34 96 586 07 37
Fax +34 96 680 23 67
The girl in the office speaks perfect english!
(The reason I would use these two before others that are recommended is that they both have offices in the Levante end of town and have both been operating for many years)