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From the documentation:

The simplex method can be applied only to problems in standard form, which can be created either directly

sage: InteractiveLPProblemStandardForm(A, b, c, ["C", "B"])
LP problem (use typeset mode to see details)

or from an already constructed problem of “general type”:

sage: P = P.standard_form()

So :

sage: foo=P.standard_form().run_simplex_method()
sage: type(foo)
<class 'sage.misc.html.HtmlFragment'>

This methods seems to be aimed at doing some computations on (supposedly toy) example(s) and "print" the wanted results, not to return any result further usable.

To get anything else, your best bet is probably to start from this method's source code ad wrap your own. Possibly following the module authors' advice :

This module, meant for educational purposes only, supports learning and exploring of the simplex method.

Do you want to solve Linear Programs efficiently? use MixedIntegerLinearProgram instead.

HTH,