To automatically spawn an ssh-agent when starting the first Bash instance, and otherwise re-register a running ssh-agent in the Windows Subsystem for Linux, append the following lines to your .bashrc. Kudos to Mathew Johnson!

# wsfl bash is not a login shell
if [ -d "$HOME/bin" ] ; then
  PATH="$HOME/bin:$PATH"
fi
 
# ssh-agent configuration
if [ -z "$(pgrep ssh-agent)" ]; then 
  rm -rf /tmp/ssh-*
  eval $(ssh-agent -s) > /dev/null
else
  export SSH_AGENT_PID=$(pgrep ssh-agent)
  export SSH_AUTH_SOCK=$(find /tmp/ssh-* -name agent.*)
fi
 
if [ "$(ssh-add -l)" == "The agent has no identities." ]; then
  ssh-add
fi