$$
\pi(\tau, \mu|\mathbf{x}) \propto \tau ^{\alpha + \frac{n}{2} - \frac{1}{2}}\text{exp}\Big(-\tau \Big[\beta + \frac{k}{2}(\mu - \nu)^2 + \frac{1}{2}\sum(x_i-\mu)^2\Big]\Big)
$$
$$
\pi(\tau, \mu|\mathbf{x}) \propto \tau ^{\alpha + \frac{n}{2} - \frac{1}{2}}\text{exp}\Big(-\tau \Big[\beta + \frac{1}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 +\frac{1}{2} \frac{nk}{n+k}(\bar{x} - \nu)^2 + \frac{1}{2}\sum(x_i - \bar{x})^2\Big]\Big)
$$
define
$$A=\frac{1}{2} \frac{nk}{n+k}(\bar{x} - \nu)^2 + \frac{1}{2}\sum(x_i - \bar{x})^2$$
$$
\pi(\tau, \mu|\mathbf{x}) \propto \tau ^{\alpha + \frac{n}{2} - \frac{1}{2}}\text{exp}\Big(-\tau \Big[\beta+A + \frac{1}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 \Big]\Big)
$$
for $\tau$
$$
\pi(\tau|\mathbf{x}) \propto \int \tau ^{\alpha + \frac{n}{2} - \frac{1}{2}}\text{exp}\Big(-\tau \Big[\beta+A + \frac{1}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 \Big]\Big) d\mu
$$
$$
\pi(\tau|\mathbf{x}) \propto \tau^{\alpha + \frac{n}{2} - \frac{1}{2}}e^{-\tau (\beta+A) }\int
e^{ -\frac{\tau}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 }
d\mu
$$
$$
\pi(\tau|\mathbf{x}) \propto \tau^{\alpha + \frac{n}{2} - \frac{1}{2}}e^{-\tau (\beta+A) }\int
e^{ -\frac{\tau}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 }
d\mu \frac{\sqrt{\frac{1}{\tau(k + n)}}}{\sqrt{\frac{1}{\tau(k + n)}}}
$$
$$
\pi(\tau|\mathbf{x}) \propto \tau^{\alpha + \frac{n}{2} - \frac{1}{2}}e^{-\tau (\beta+A) }
\sqrt{\frac{1}{\tau(k + n)}}\int \frac{1}{\sqrt{\frac{1}{\tau(k + n)}}}
e^{ -\frac{\tau}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 }
d\mu
$$
do you see that integral $\propto 1$??(normal distribution).
$$
\pi(\tau, \mu|\mathbf{x}) \propto \tau ^{\alpha + \frac{n}{2} - \frac{1}{2}}\text{exp}\Big(-\tau \Big[\beta + \frac{1}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 +\frac{1}{2} \frac{nk}{n+k}(\bar{x} - \nu)^2 + \frac{1}{2}\sum(x_i - \bar{x})^2\Big]\Big)
$$
$$
\pi( \mu|\tau,\mathbf{x}) \propto \text{exp}\Big(-\tau \Big[ \frac{1}{2}(k + n)\Big(\mu-\frac{k\nu + n\bar{x}}{k + n}\Big)^2 \Big]\Big)
$$