How do I get the voltage of Ro to calculate the maximum power transfer?
Re-drawn, your schematic looks about like this:



simulate this circuit &ndash; Schematic created using CircuitLab
I made a choice about ground. You get to pick exactly one node and call it ground. Since this is a maximum-power question, I wanted to pick one side of \$R_O\$ and call that ground, so that I could just focus on the output impedance as seen from the other side. Single-ended analysis is easier.
I've also left \$R_O\$ disconnected, but pointing to where it will be connected, to highlight that this is a maximum-power question. You probably already know that the maximum power will be obtained when \$R_O\$ is the same resistance as the output resistance of the circuit without \$R_O\$ attached across \$R_2\$.
In laying out the schematic I've kind of assumed, without knowing for sure, that \$V_1\$ and \$V_2\$ are probably negative values. And since I want positive at top and negative at bottom, I've arranged things according to my guess.
Finally, there's no need at all to show where that voltage-dependent current source sinks. It sinks to a known voltage reference and it's entirely irrelevant to the circuit. Besides, current sources/sinks have infinite impedance (open circuit, basically.) So I just left one end hanging. It's simpler to consider and doesn't alter a single thing.
Once you are at this point, you just have two very simple nodal equations to develop. (For \$V_1\$ and for \$V_2\$.) You will want to analyze the above circuit to get the node voltage for \$V_2\$. Then you will want to analyze the circuit a second time, but looking like this:



simulate this circuit
By adding a one amp current into the \$V_2\$ node, you will see that its resulting voltage changes, as a result of it. The change in voltage, the difference between the two solutions for \$V_2\$, is then divided by the current change (which is obvious), and that tells you the Thevenin source impedance for the entire circuit -- absent \$R_O\$, of course.
That source impedance will then also be the impedance you want for \$R_O\$ to obtain the maximum power in \$R_O\$ for this circuit.
(That value, I can assure you, is not \$\approx 1.674\:\Omega\$.)