In the diagram, QT ext || WU ext and QR ext || WT. Find the length of ST.

Given that $QR \parallel WT$.
Consider $\triangle QRS$ and $\triangle WTS$.
Since $WT \parallel QR$, we have:
$\angle S$ is common to both triangles.
$\angle STW = \angle SRQ$ (corresponding angles).
Therefore, $\triangle WTS \sim \triangle QRS$ by AA similarity.

From the similarity, the ratio of corresponding sides is equal:

$$\frac{WS}{QS} = \frac{ST}{SR}$$

We are given $QW = 12 \text{ mm}$ and $WS = 11 \text{ mm}$.
The length of $QS$ is $QW + WS = 12 + 11 = 23 \text{ mm}$.
Substitute the known values into the ratio:

$$\frac{11}{23} = \frac{ST}{SR} \quad (*)$$

Given that $QT \parallel WU$.
Consider $\triangle QST$ and $\triangle WSU$.
Since $WU \parallel QT$, we have:
$\angle S$ is common to both triangles.
$\angle SWU = \angle SQT$ (corresponding angles).
Therefore, $\triangle WSU \sim \triangle QST$ by AA similarity.

From the similarity, the ratio of corresponding sides is equal:

$$\frac{SW}{SQ} = \frac{SU}{ST}$$

We know $SW = 11 \text{ mm}$ and $SQ = 23 \text{ mm}$.
Substitute these values into the ratio:

$$\frac{11}{23} = \frac{SU}{ST}$$

From the ratio in Step 2, we can express $SU$ in terms of $ST$:

$$SU = \frac{11}{23} ST$$

We are given $TU = 6 \text{ mm}$.
From the diagram, points $S, U, T$ are collinear on $RS$ in that order, so $ST = SU + UT$.
Substitute the expression for $SU$ and the value of $TU$:

$$ST = \frac{11}{23} ST + 6$$

Subtract $\frac{11}{23} ST$ from both sides:

$$ST - \frac{11}{23} ST = 6$$ $$\frac{23 ST - 11 ST}{23} = 6$$ $$\frac{12 ST}{23} = 6$$

Multiply both sides by 23 and divide by 12:

$$12 ST = 6 \times 23$$ $$12 ST = 138$$ $$ST = \frac{138}{12}$$ $$ST = 11.5 \text{ mm}$$

Substitute the value of $ST = 11.5 \text{ mm}$ into equation $(*)$ from Step 1:

$$\frac{11}{23} = \frac{11.5}{SR}$$

Multiply both sides by $23 \times SR$:

$$11 \times SR = 23 \times 11.5$$ $$11 SR = 264.5$$

Divide by 11:

$$SR = \frac{264.5}{11}$$ $$SR = 24.04545... \text{ mm}$$

Rounding 24.04545... to the nearest whole number gives 24.

$$RS \approx 24 \text{ mm}$$

The final answer is $\text{24 mm}$.