After wasting 2-3 days trying to get Live Writer working after updating to WO 2.3.1, I finally stumbled across a solution that worked and had not seen posted anywhere else.
WP's xmlrpc.php calls a file class-IXR.php in the wp-includes folder. The problem I have is that my hosting server (GoDaddy) is case sensitive. and the file had the name class-ixr.php. After I changed the filename to match the xmlrpc call, no more problems (so far.)
Just something to look at so you do not tear your hair out like I did.