How can I mirror a group of components and wires across a line in eagle, without having it switch board sides?
You can't do that.

At best you could position the components on the right side symmetrically about the cyan line relative to their counterparts on the left.  However, the components themselves can't be mirrored.  This only works for symmetrical parts, like resistors for example.  It's not going to work for whatever the 8-pin chip is since you can't magically mirror its pinout.