Normally I just take $q(M_{m\times n} - I_{m\times n})$ to workout the steady state, but here I have:
$$\left(\begin{array}{rrrr} 0 & 0 & .8 & .2 \\ .4 & .6 & 0 & 0 \\ .2 & .8 & 0 & 0 \\ 0 & 0 & .7 & .3\end{array}\right).$$
I can't use my normal method for it would be excessive for this matrix, how can I solve problems like this with $4 \times 4$ transition matrices
edit: If I know $\lim \limits_{n \to \infty} P^n$, can I use their values somehow?
You should be able to solve it your way. An equivalent way to solve it is by
$ V = V*\left[ \begin{array}{cccc} 0 & 0 & .8 & .2 \\ .4 & .6 & 0 & 0 \\ .2 & .8 & 0 & 0\\ 0 & 0 & .7& .3\end{array} \right] $, where $V = \left[ \begin{array}{cccc}a & b & c & d \end{array} \right] $ and then solving the resulting equations for $a, b, c, d$, with the fact that $a + b + c + d = 1$.
If you'll settle for a numerical answer you can always code this up. Ether way, the answer is $V = \left[ \begin{array}{cccc}.2333 & .4667 & .2333 & .0667 \end{array} \right]$.
Also, you can numerically find that $\lim_{n \rightarrow \infty}P^n = \left[ \begin{array}{cccc} .2333 & .4667 & .2333 & .0667 \\ .2333 & .4667 & .2333 & .0667 \\ .2333 & .4667 & .2333 & .0667\\ .2333 & .4667 & .2333 & .0667\end{array} \right] $