For future searchers…
It seems the WD only offers a low-security SSH connection. I was using mintty, which uses OpenSSH. They have disabled DSA keys: http://www.openssh.com/legacy.html That page explains why, and what to do about it.
OpenSSH 7.0 and greater similarly disables the ssh-dss (DSA) public key algorithm. It too is weak and we recommend against its use. It can be re-enabled using the HostkeyAlgorithms configuration option:
ssh -oHostKeyAlgorithms=+ssh-dss firstname.lastname@example.org
I got in using PuTTY, and based on that discovered the above solution. That option allowed me to do SSH using my normal tools.
Looks like it’s time for an upgrade to WD’s SSH.