Exhibit a $13$-digit integer $N$ that is an integer multiple of $2^{13}$ and whose digits consist of only $8$s and $9$s.
To find a 13-digit integer \( N \) that is an integer multiple of \( 2^{13} \) and whose digits consist only of \( 8 \)s and \( 9 \)s, we need to ensure that \( N \) is divisible by \( 8192 \) (since \( 2^{13} = 8192 \)).

1. **Last Digit Analysis**:
   - For \( N \) to be divisible by \( 2 \), its last digit must be even. Therefore, the last digit must be \( 8 \).

2. **Last Two Digits Analysis**:
   - For \( N \) to be divisible by \( 4 \), its last two digits must form a number divisible by \( 4 \). The possible combinations are \( 88 \) and \( 98 \). Only \( 88 \) is divisible by \( 4 \).

3. **Last Three Digits Analysis**:
   - For \( N \) to be divisible by \( 8 \), its last three digits must form a number divisible by \( 8 \). The possible combinations are \( 888 \) and \( 988 \). Only \( 888 \) is divisible by \( 8 \).

4. **General Pattern**:
   - Continuing this pattern, we see that for \( N \) to be divisible by \( 2^n \) for \( n \leq 13 \), the last \( n \) digits must form a number divisible by \( 2^n \). This pattern suggests that the last \( n \) digits must be \( 888 \ldots 8 \) (with \( n \) digits of \( 8 \)).

5. **Verification**:
   - We need to verify if \( 8888888888888 \) (13 digits of \( 8 \)) is divisible by \( 8192 \).

\[
8888888888888 \div 8192 = 1085276
\]

   - Since the division results in an integer, \( 8888888888888 \) is indeed divisible by \( 8192 \).

Therefore, the 13-digit integer \( N \) that is an integer multiple of \( 2^{13} \) and whose digits consist only of \( 8 \)s and \( 9 \)s is \( 8888888888888 \).

The final answer is \(\boxed{8888888888888}\).