How can I convert Chinese characters sketched with my mouse, into computerised characters? Are there any websites or resources analogous to Detextify for maths?
Presently, I do not know (but desire to learn) how to type Chinese characters.
There are many online OCR input websites. For example:

http://www.yibizi.com/html5/ (neat and fast, without ads, but seems only working inside China)

I strongly suggest you install a client software rather than using a website because many websites are full of ads. And many of them are only working in some particular browsers, like infamous IE series.

Sougou Pinyin (wikipedia page) is the most used Input Method in China. I am also personally using this on my MacBook Pro. It has Windows/Mac version. You can get the software by clicking the links in the Wikipedia page.

With it, you can click "手写输入" to input by "drawing":