Can someone explain to me how to apply Kirchoff's Law to this circuit?
I might be a little rusty, but here goes.
You write a set of equations with Kirchoff's Current Law (KCL) and/or
Kirchoff's Voltage Law (KVL):
The sum of the currents entering or leaving any node is zero. The sum
of the voltages around any loop is zero.
Then you solve the set of simultaneous equations.
In your circuit, there are two nodes, upper and lower, with a junction
of three resistors at each node. If you (arbitrarily) define the
directions of the currents in the resistors R1-R4 to be INTO their
respective nodes, and the direction of the current in R5 to be "up"
(i.e. into the upper node), then KCL would give you:
R4 I4 + R5 I5 + R3 I3 = 0
R1 I1 + R2 I2 - R5 I5 = 0 .
And, proceeding counterclockwise around each of the two inner loops,
left loop first, KVL would give you:
6 - R1 I1 - R5 I5 + R4 I4 = 0
6 - R3 I3 + R5 I5 - R2 I2 = 0
That's four equations for five variables. One more equation is needed.
The get a fifth equation, applying KVL to the outer loop gives you:
6 + R1 I1 - R2 I2 + 6 + R3 I3 - R4 I4 = 0
Check my work. I may have blundered on the signs, or something.
If you can't find a simpler solution, then you could form these five
equations into one multi-dimensional equation and use Linear Algebra to
solve it. i.e. [i vector] x [R matrix] = [constant vector], then [i
vector] = [constant vector] x [inverted R matrix].