How do I determine the reaction speed?
The rate of reaction can be determined experimentally or theoretically, the latter case requiring that a reaction mechanism be known. Proposing a reaction mechanism requires knowledge of the reaction at multiple regimes and some guesswork. Evidence for a certain reaction mechanism is usually indirect.
Theoretically speaking, once the reaction mechanism and rate constants are known, we can write down a coupled system of kinetic rate equations and solve them simultaneously to find the rate of reaction.
I know you need a value, k, and order, which is only gained through experimentation, but how do the differential equations play in? How do I plot the result?
The differential equations govern the reacting system, and their solution completely specifies the system, enabling you to find the rate of reaction. You would then simply plot, say, the rate of reaction vs. time.
How do I proceed? I now have this function but I am not sure what to do with it. I do not have the means to observe the reaction and figure out its k-value or order.
I don't believe you can proceed, because your initial assumption was wrong. The combustion of hydrogen is far more complicated than a first-order reaction, which you have assumed; you should not, therefore, expect that your first-order analysis should describe your system well.
Is this reaction a known one, though? Is the order and k-value something you can find on the Internet?
It has certainly been studied. I will refer you to Reaction mechanism of combustion of H2 and the links therein.