Senate has no mechanism to prosecute contempt charges

That goes to the DOJ, which of course means nothing happens, at least for next 800 days

It's my understanding they can detain under the inherent contempt power, it's been reported [0] that the House has a jail that can be readied if needed.

[0] https://rollcall.com/2019/05/08/just-where-is-this-secret-ho...