Just about every street in Santa Eulalia has either an internet cafe or a bar with free wi-fi as long as you buy a coffee. Cafe Sydney and Rumours in the marina both offer it. If you don't have your own computer look for signs saying Locutorio, Mark & Dannis (a computer shop just off Cannon Square) is good and they speak English.
I know of four launderettes off the top of my head, there is one on the road leading down to Dona Ana a few doors down from the tabac, one on the road out to Es Cana on the right hand side just before the car wash, one opposite Cana Suni and one in the marina behind Sa Llesca. None of them are self service though so you have to have a service wash. 5kgs including ironing should cost between €16 and €20.