Would some help me fill in missing steps from a textbook exercise left for the reader
Overall it is almost correct but there are some small mistakes and it is incomplete.
1st Mistake:
in step (1) you write:

$\frac{Y_t}{L_t}=\frac{K^{\alpha}_t}{L_t}\frac{L^{1-\alpha}_t}{L_t}$

But it should actually be
$$\frac{Y_t}{L_t}=\frac{K^{\alpha}_t L^{1-\alpha}_t}{L_t}$$
You cannot divide LHS of the equation by $L$ and RHS by $L L \implies L^2$ as that will generally violate the equality between $ Y= K^{\alpha}L^{1-\alpha}$.
However, this error had no impact on the rest of the problem because in spite of making this mistake you still used the correct definition for output per capita:  $y_t=k^{\alpha}_t$
2nd mistake
you say you divide both sides by $k_t$, but you forgot to do that for the last term. You write:

$0 = \frac{\phi[k^{\alpha}_t\sigma]}{k_t}+\frac{sk^{\alpha}_t}{k_t}+\frac{\lambda[k^{\alpha}_t\sigma]}{k_t}-(\delta+n)k_t$

correct steady state equation should be:
$$0 = \frac{\phi[k^{\alpha}_t\sigma]}{k_t}+\frac{sk^{\alpha}_t}{k_t}+\frac{\lambda[k^{\alpha}_t\sigma]}{k_t}-(\delta+n) \tag{*}$$
3rd 'mistake': It is unfinished
I am using here ‘mistake’ in quotes because this is not a mistake in the same sense as other ones but it is still not good practice to leave problem unfinished. If you would do it on an exam it would cost you some points (some professors might even refuse to give any points for unfinished work).
I get the impulse to just call it a day since you are in home stretch now but proper answer should have solution for $k^*$ (unless the problem specifically states it does not want full solution which I don’t see in this case in your Q).
Solving the equation $*$ for optimal $k^*$ yields:
$$ k^* =  \left( \frac{(\phi+\lambda)\sigma +s}{\delta +n} \right)^{\frac{1}{1-\delta}}$$
The above being said don’t get discouraged most of the attempt was correct. Just be more careful about the arithmetic and don’t stop just before finishing.