Now that this answer is old it is worth providing a full solution. Let's start by looking at the maximised log-likelihood for data $Y_1,...,Y_n \sim \text{IID Pois}(\lambda)$. The log-likelihood function for this case is given by:
$$\ell_\mathbf{y}(\lambda) = n \Big[ \bar{y} \ln (\lambda) - \lambda \Big] + \text{const}
\quad \quad \quad \text{for } \lambda \geqslant 0.$$
The corresponding score function and information function are given respectively by:
$$\begin{aligned}
s_\mathbf{y}(\lambda) &= n \Big[ \frac{\bar{y}}{\lambda} - 1 \Big], \\[6pt]
I_\mathbf{y}(\lambda) &= \frac{n \bar{y}}{\lambda^2}. \\[6pt]
\end{aligned}$$
Since the information is strictly positive, the log-likelihood is strictly concave, and so the MLE occurs at the unique critical point where $s_\mathbf{y}(\hat{\lambda})=0$, which gives $\hat{\lambda} = \bar{y}$. Thus, the maximised log-likelihood is:
$$\hat{\ell}_\mathbf{y}
\equiv \max_{\lambda} \ell_\mathbf{y}(\lambda)
= \ell_\mathbf{y}(\hat{\lambda})
= n \bar{y} [\ln (\bar{y}) - 1] - \sum_{i=1}^n \ln(y!).$$
This gives the general form of the maximised log-likelihood function. In the present problem we have two data points that are either IID (under the null hypothesis) or they are independent values that are not identically distributed (under the alternative hypothesis). The resulting maximised log-likelihood functions under the two models are as follows.
Null hypothesis: Under the null hypothesis $H_0: \lambda_1 = \lambda_2$ we have:
$$\hat{\ell}_0 = (y_1+y_2) \ln \Big( \frac{y_1+y_2}{2} \Big) - (y_1+y_2) - \ln(y_1!) - \ln(y_2!).$$
Alternative hypothesis: Under the alternative hypothesis $H_A: \lambda_1 \neq \lambda_2$ we have:
$$\hat{\ell}_A = y_1 \ln (y_1) + y_2 \ln (y_1) - (y_1+y_2) - \ln(y_1!) - \ln(y_2!).$$
Likelihood ratio statistic: The above forms for the maximised log-likelihood functions give the likelihood ratio statistic:
$$\begin{aligned}
\Delta (y_1, y_2)
&= 2(\hat{\ell}_A - \hat{\ell}_0) \\[6pt]
&= 2 \Bigg[ y_1 \ln (y_1) + y_2 \ln (y_1) - (y_1+y_2) \ln \Big( \frac{y_1+y_2}{2} \Big) \Bigg] \\[6pt]
&= 2 \sum_{i=1}^2 y_i [ \ln (y_1) - \ln (\bar{y}) ] \\[6pt]
&= 2 \sum_{i=1}^2 y_i \ln \Big( \frac{y_i}{\bar{y}} \Big). \\[6pt]
\end{aligned}$$
self-studytag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. $\endgroup$