DPDA for $\{1^ky \mid \text{$y\in \{0,1\}^*$ with $|y|_1 \le k$ and $k \in \mathbb N: k\ge1$}\}$
In order to verify whether the string is of the form $1^ky$ with $y\in \{0,1\}^*$ and $|y|_1\le k$ you want to determine whether a symbol $1$ in the prefix belongs to the $1^k$ part, or starts $y$. 

Now imagine $1^k \cdot 1y$ is of the correct form, i.e., $|1y|_1 \le k$, then certainly $1^{k+1}\cdot y$ is also correct, as $|y|_1&lt; |1y|_1 \le k &lt; (k+1)$.

So, keep reading $1$'s while possible.