Speaking from experience of sponsoring UK visitor visas for family members many times over the last 10 years or so, I can confidently say that typed letter is perfectly fine. Naturally, it should have your signature at the bottom, signed by hand (although on several occasions a scanned/printed copy was accepted, which obviously did not contain the original signature).
Depending on which country you are applying from, 2-3 weeks is absolutely normal. In fact, even a month is not too abnormal. Last time I sponsored a visitor visa for a relative was only a few months ago (January 2017) - and it took about 3 weeks from application to the decision.
Just relax and don't worry about things like this. Provided everything else is in order, the visa should be issued.
UPDATE: It's worth emphasising that the sponsorship letter, just like all the rest of the supporting documentation, must be in English.