$$T=f(t):=2 \alpha + \sqrt{\alpha^2+t^2}$$
To estimate $\alpha$ we got the measured values $T_i$ for $t_i$. Formulate the curve fitting problem and show each step in the Gauss-Newton algorithm.
My Idea: The Problem $$\|T(\alpha^*)\| = \min\|T(\alpha)\| \text{ and } \|T(\alpha)\|= \sum f(t_i) -T_i$$
Now to get the right $\alpha$ i have to do the iteration $$\alpha^{k+1}=\alpha^k+r^k$$
$T'$ is the derivative and $T^T$ is the Transpose $$ r = -(T'(\alpha)^T * T'(\alpha))^{-1}*T'(\alpha)^T *T(\alpha)\\ T'(\alpha) = 2+\frac{\alpha}{\sqrt{\alpha^2+t_i^2}} $$
Now I put the Terms $T,T'$ in the Equation and I am done. That's how I would do it. But I am not sure if I have to check some properties of the function $f(t)$ or if it's really the right way.
Your notations are little bit wonky. But, basically you got it.
Gauß-Newton is about minimizing the squared norm $\| F(\alpha) \|^2 = \sum_{i=1}^n (F_i(\alpha))^2$ of a vector valued function, say $F:\mathbb R\to\mathbb R^n$. Here, $$ F_i(\alpha) = 2\alpha + \sqrt{\alpha^2 + t_i^2} - T_i $$ for $1\le i\le n$.
To compute the Gauß-Newton step $r$ at $\alpha$, you solve the linear equation (here only 1D) $$ F'(\alpha)^T F'(\alpha) r = - F'(\alpha)^T F(\alpha), $$ that is $$ r = - \frac{F'(\alpha)^T F(\alpha)}{F'(\alpha)^T F'(\alpha)} = -\frac{\sum_{i=1}^n \bigl(F'_i(\alpha)\bigr)^2}{\sum_{i=1}^n F'_i(\alpha) F_i(\alpha)} $$ with $$ F_i'(\alpha) = 2+\frac{\alpha}{\sqrt{\alpha^2 + t_i^2}}. $$