Let the two unknown observations be $a,b$. Sum of the 8 known = $80$.

Mean condition: $\frac{80+a+b}{10}=9\Rightarrow a+b=10$. ...(1)

Variance condition: $\frac{\sum x_i^2}{10}-81=34.2\Rightarrow \sum x_i^2=1152$.

${\sum }_{\text{known}}{x}_i^2=4+9+25+100+121+169+225+441=1094$, so $a^2+b^2=58$. ...(2)

From (1) and (2): $(a+b{)}^2-2ab=58\Rightarrow 100-2ab=58\Rightarrow ab=21$. Solving with $a+b=10$: $\{a,b\}=\{3,7\}$.

Sorted data: $2,3,3,5,7,10,11,13,15,21$.

Median (average of 5th and 6th) $=\frac{7+10}{2}=8.5$.

Mean deviation about median: $\frac{|2-8.5|+|3-8.5|+|3-8.5|+|5-8.5|+|7-8.5|+|10-8.5|+|11-8.5|+|13-8.5|+|15-8.5|+|21-8.5|}{10}$

$=\frac{6.5+5.5+5.5+3.5+1.5+1.5+2.5+4.5+6.5+12.5}{10}=\frac{50}{10}=5$.

Option (1).