That doesn't prove the letter to be authentic.Good question. Much of what happened to him after his arrest is obscure and unknown. In the past six decades the Governments both Congress and non-Congress ones have systematically tried to censor information about Godse. It is not specifically known when exactly this letter was leaked by which exact book. However you can take this letter to be authentic because the same has been cited in various websites, some which praise Gandhi.
However, it is in our interest to let the truth be known and you are encouraged to disprove the authenticity of this letter if it is indeed so.
I'm no fan of Gandhi, nor Ghodse; so either way, I'm content with knowing the truth. However, citing Internet sites, copy pasting each other, does not prove authenticity!