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: OpenSSH: Legacy Options 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 user@127.0.0.1
I got in using PuTTY, and based on that discovered the above solution. That option allowed me to do SSH using my normal tools.