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Let $\angle SPT = \angle TPU = \theta$ . Then $\angle UPQ = 90^\circ - 2\theta$ , and

$\begin{aligned} |UR| & = |RQ| - |PQ|\tan \angle UPQ \\ & = 2 - 2 \tan (90^\circ - 2\theta) \\ & = 2 - 2\cot (2\theta) \\ & = 2 - 2 \times \frac {1-\tan^2 \theta}{2\tan \theta} & \small \blue{\text{Note that }\tan \theta = \frac 12} \\ & = 2 - 2\times \frac {1-\frac 14}{2 \times \frac 12} \\ & = \frac 12 = \boxed{0.5} \end{aligned}$

Let $\angle {SPT}=\angle {TPU}=α$ . Then $\tan α=\dfrac{1}{2}, \tan (2α)=\dfrac{2}{2-|\overline {UR}|}=\dfrac{2\times \frac{1}{2}}{1-\frac{1}{4}}=\dfrac{4}{3}\implies |\overline {UR}|=\dfrac{1}{2}=\boxed {0.5}$ .