Step 1: Understand the problem scope.
We need to count all numbers strictly between 10 and 10000 formed from digits $\{1,2,3,4,5\}$ with no repetition. Numbers in this range are 2-digit, 3-digit, and 4-digit numbers. Since no digit is 0, there is no restriction on the leading digit.
Step 2: Count 2-digit numbers.
Choose and arrange 2 digits from 5 available digits (order matters, no repetition). This is the permutation ${}^5P_2 = 5 \times 4 = 20$.
Step 3: Count 3-digit numbers.
Choose and arrange 3 digits from 5 available digits. This is ${}^5P_3 = 5 \times 4 \times 3 = 60$.
Step 4: Count 4-digit numbers.
Choose and arrange 4 digits from 5 available digits. This is ${}^5P_4 = 5 \times 4 \times 3 \times 2 = 120$.
Step 5: Why not 5-digit numbers?
The upper bound is 10000 (exclusive). A 5-digit number formed from $\{1,2,3,4,5\}$ would be at least 12345, which exceeds 10000, so no 5-digit numbers qualify.
Step 6: Add all cases.
Total = $20 + 60 + 120 = 200$.
\[ \boxed{200} \]