How to derive the formula for heat produced due to electricity correctly from Joule's laws for heating?

From [Joule's laws][1], we get this:

$$Hpropto I^{2}Rt$$

Just to be clear, since your post is under the thermodynamics tag, you should be aware that the left side of your equation is technically not "heat". In thermodynamics heat is defined as energy transfer due solely to temperature difference. In this case the energy transfer is electrical work per unit time (power) that is converted to internal molecular kinetic energy of the resistor. The increase in resistor temperature can (but not necessarily) result in heat transfer to the surroundings by conduction, convection and/or radiation, depending on the nature of the surroundings. For this reason in electrical engineering it is referred to as resistance "heating".

$$implies H = KI^{2}Rt ...(i)$$

Now, we have to find/define the value of K. According to my book, when $1A$ of current passes through a conductor of $1Omega$ for $1s$, $1J$ heat is produced. If that's the case, then from $(i)$ we get this: $$1=Ktimes1times1times1$$ $$implies K = 1$$

Actually, if you do say the left side of the equation is "heat", then $K$ will be something less than 1. That's because part of the power delivered to the resistor goes into increasing its internal energy (increasing the temperature of the resistor) and the rest gets transferred to the surroundings in the form of heat due to the temperature difference between the resistor and its surroundings.

My question with the above derivation is, how did we find that when $1A$ of current passes through a conductor of $1Omega$ for $1s$, $1J$ heat is produced? Through experimentation? What was the name of the experiment and who conducted it?

I don't recall the details of how it was done historically, but one method might be calorimetry. Immerse an electrical heater in a known amount of water and known temperature contained in a thermally insulated vessel. Operate the heater at known power for a known time. That will give you the Joules delivered to the water. Measure the increase in water temperature and using the specific heat of water calculate the amount of Joules of energy the water received. The two values should be close, within the margins of error of the experiment.

Hope this helps.

Another derivation goes like this:

consider a resistance $R$ connected to an ideal battery of emf $E$. acc. to Ohm's law,$$E=Icdot R$$ where $I$ is the current in the circuit.

in time $dt$ a charge $Icdot dt$ passes through the battery, hence work done by the battery is $dW=EIdt$. This work done appears as heat in the resistance and hence, $$dH=EIdt=(IR)Idt=I^2Rdt$$ on integrating, $$H=I^2Rt$$