The card after the first ace

1. Step 1 · Take the king of hearts out

   $\displaystyle \text{shuffle the other } 51 \text{ cards}$

   Build the shuffle in two steps: shuffle the other 51 cards, then put the king of hearts back into a random gap.

2. Step 2 · Gaps it could go back into

   $\displaystyle \text{before},\, \text{ between or after the } 51 \text{ cards } = 52$

   Each of the 52 gaps is equally likely, and the order of the other cards is untouched.

3. Step 3 · Gaps that put it right after the first ace

   $\displaystyle \text{exactly } 1$

   Whatever the first ace is, exactly one gap sits right after it. So the chance is the same as for any one card.

4. Step 4 · Chance

   $\displaystyle 1 \div 52 = \frac{1}{52}$

   One good gap out of 52 equally likely ones.

Answer$\displaystyle \frac{1}{52} \approx 1.92\%$