There is a decent laundrette called Sol near city centre, behind Radisson Blu Hostel. Wash and dry is 6 Euros.
www.sol.ee
Or try one of the youth hostels. Though use at your own risk.
It is a human right in Estonia to have internet access. So free wifi for all pretty much everywhere.
AIESECers recommend:
"If you want to have a quiet place away from all the everyday noise, you can visit the Estonian center library (next to freedom square).
You don´t need to sign up to visit and you can read there or take your laptop with you (free wifi)."
Although no one checks when you board public transport. Do make sure that you have a valid ticket or touch your Green Card on the reader. There are ticket inspectors.
Night View from Lounge 24