Practice is the best way to improve writing speed. One technique is to set aside a period of time each day for speed writing. The object is to write at top speeds, without stopping for any reason, until the time is up, without worrying about grammar, spelling or even content.
After a few of these sessions, the writer reviews the work, identifies recurring errors, and creates of list of areas to work on until the correct spelling or grammatical constructs become automatic. Another advantage of this technique is that by not judging the content, the inner critic is silenced, allowing for more creativity.