= Solution
Put $\beta=m\alpha/\hbar^2$. The delta interaction reduces the <integral equation> to $\psi_s=e^{ikx}+i\beta e^{ik|x|}\psi_s(0)/k$. At zero, $\psi_s(0)=(1-i\beta/k)^{-1}$. Hence
$$
\boxed{t=\frac{k}{k-i\beta},\qquad r=\frac{i\beta}{k-i\beta},\qquad T=\frac{k^2}{k^2+\beta^2},\quad R=\frac{\beta^2}{k^2+\beta^2}.}
$$
Their sum is one, as required for a real potential. The explicit <wave> is the indicated incident-plus-reflected form for $x<0$ and $t e^{ikx}$ for $x>0$; it is continuous and has <derivative> jump $-2\beta\psi(0)$.
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