$$ y(t)=\mathcal{L}^{-1}\left[\frac{K}{s(T s+1)}\right] $$
I know it should be: $$ K\left(1-e^{-t / T}\right) $$
But I get:
$$ y(t)=\mathcal{L}^{-1}\left[\frac{K}{s(T s+1)}\right] = [\text{partial fraction division}] = K \left( \mathcal{L}^{-1}\frac{1}{s} + \mathcal{L}^{-1}\frac{1-T}{sT+1} \right) = K\left( 1 + \frac{1}{T} \frac{1}{s + \frac{1}{T}} - \mathcal{L}^{-1}\frac{1}{T} T \frac{1}{s + \frac{1}{T}} \right)= K \left( 1 + \frac{1}{T}e^{-1/T} - e^{-1/T} \right)= K \left( 1 + \frac{T-1}{T}e^{-1/T} \right) $$
Where did it go wrong?
Your partial fraction decomposition is wrong. Note that $$\frac{K}{s(T s+1)} = \frac{K}{s} - \frac{KT}{sT+1} = \frac{K}{s} - \frac{K}{s+\frac{1}{T}} $$Now take the inverse transform which yields $$y(t) = Ku(t) - Ke^{-\frac{t}{T}}u(t) = Ku(t)(1-e^{-\frac{t}{T}})$$