Thermal noise (Johnson noise)
Your problem is combining the voltage sources. This is incorrect, first because you can't add uncorrelated noise to each other, second because we don't even need to do worry about the other resistor's power generation for this problem.

Since we are only looking at the power that one resistor transfers to another, we look only at the voltage it generates and transfers to the other.







simulate this circuit &ndash; Schematic created using CircuitLab

Now, we look at the voltage that would appear on the transferred resistor, which would be exactly half.

$$
V_\mathrm{transferred} = \frac{\sqrt{4k_BT \Delta FR}}{2}
$$

Now with power:
$$
P_\mathrm{transferred} = \frac{V^2}{R}
$$
$$
P_\mathrm{transferred} = \frac{4k_BT \Delta FR}{4R}
$$
$$
P_\mathrm{transferred} = k_BT \Delta F
$$

Hope this helps!