Let $x$ be the independent variable and $y$ the dependent variable, a lineal relation between this variables is given by $y_i = \beta_0 + \beta_1 x_i+ u_i$. If the estimator of $\beta_0$ by OLS is defined by $\hat{\beta_0} = \bar{x} - \hat{\beta_1}\bar{y}$ and $\hat{\beta_0}=\sum_{i=1}^{T}(\frac{1}{T}-\bar{x}k_i)y_i$ where $k_i=\frac{x_i-\bar{x}}{\sum_{i=1}^{T}(x_i-\bar{x})^2}$. Prove that $$Var(\hat{\beta_0})=\hat{\sigma_2}\left[\frac{1} {T}+ \frac{\bar{x}^2}{\sum_{i=1}^{T}(x_i-\bar{x})^2}\right]$$
This is what I've tried: $$Var(\hat{\beta_0})=\mathbb{E}\left[\hat{\beta_0}-\mathbb{E}(\hat{\beta_0})^2\right]=\mathbb{E}[\hat{\beta_0}-\beta_0]^2=\mathbb{E}\left[\left(\sum_{i=1}^{T}(\frac{1}{T}-\bar{x}k_i)y_i\right)-\beta_0\right]^2$$