Soddy-Gosset comes to the help

As we can see here , Descartes' Circle Theorem can be generalised to $n$ dimensions. This is Soddy-Gosset theorem:

In n-dimensional Euclidean space, the maximum number of mutually tangent $(n - 1)$ -spheres is $n + 2$ . The curvatures ${{k}_{i}}$ of these hyperspheres satisfy the relation $n\left( \sum\limits_{i=1}^{n+2}{{{k}_{i}}^{2}} \right)={{\left( \sum\limits_{i=1}^{n+2}{{{k}_{i}}} \right)}^{2}}$

We can consider the plane as a sphere of infinite radius (i.e. 0 curvature) and simply apply Soddy-Gosset theorem for $n=3$ :

$\begin{aligned} 3\left( \sum\limits_{i=1}^{5}{{{k}_{i}}^{2}} \right)={{\left( \sum\limits_{i=1}^{5}{{{k}_{i}}} \right)}^{2}} & \Rightarrow 3\left\{ {{\left( \frac{1}{5} \right)}^{2}}+{{\left( \frac{1}{7} \right)}^{2}}+{{\left( \frac{1}{10} \right)}^{2}}+{{\left( 0 \right)}^{2}}+{{\left( \frac{1}{r} \right)}^{2}} \right\}={{\left( \frac{1}{5}+\frac{1}{7}+\frac{1}{10}+0+\frac{1}{r} \right)}^{2}} \\ & \Leftrightarrow 3\left( \frac{69}{980}+\frac{1}{{{r}^{2}}} \right)={{\left( \frac{31}{70}+\frac{1}{r} \right)}^{2}} \\ & \Leftrightarrow 37{{r}^{2}}-2170r+4900=0 \\ & \Leftrightarrow r\approx 2.3524\text{ or }r\approx 56.296 \\ \end{aligned}$ Accepted solution for our tiny sphere, $r\approx \boxed{2.3524}$ .

(Hosam, the title of your problem is a real spoiler!)