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Not clear to me what the problem may be. To help others explore, here is a slightly edited version of the original poster's code.

<h2>Double integrator</h2>
<p>Alternative to Mathematica's double integrator. Way better!</p>
<div id="mycell">
x, y = var('x y')
html("<h1>Double Integrator</h1>")
@interact
def interplay(function=input_box(x*y),
              lower_x_bound=input_box(0),
              upper_x_bound=input_box(x),
              lower_y_bound=input_box(0),
              upper_y_bound=input_box(1),
              dydx=checkbox(default=False)):
  try:
    if dydx is False:
      result = integral(integral(function, x, lower_x_bound, upper_x_bound),
                        y, lower_y_bound, upper_y_bound)
      q = r"$\int_{%s}^{%s} \int_{%s}^{%s} %s \,dx\,dy = %s $" % (
            lower_y_bound, upper_y_bound,
            lower_x_bound, upper_x_bound,
            function, result)
    else:
      result = integral(integral(function, y, lower_y_bound, upper_y_bound),
                        x, lower_x_bound, upper_x_bound)
      q = r"$\int_{%s}^{%s} \int_{%s}^{%s} %s \,dx\,dy = %s $" % (
            lower_y_bound, upper_y_bound,
            lower_x_bound, upper_x_bound,
            function, result)
    pretty_print(html("%s" % q))
  except Exception as e :
    print("Please enter better inputs")
    print(e)
html("<p>Done</p>")
</div>