I had run into a similar issue running 10.2.1. My work around was to create the folder it was wanting to write to in my case RAS43 in the system32 folder. I also shortened the name from LongestFlowPath to LFP or similar. Not sure which one did it, but ran it again and all is well. I've noticed over the years that shortening names, not using long file paths, not using special characters or spaces, making sure everything is in the same projection, and making sure your DEM covers your area of interest cures just about 99% of ArcHydro issues. Hope this helps.
... View more