Making nitric acid from KNO3 and H2SO4 is perfectly possible. The only thing is that you need all-glass apparatus. HNO3 is very corrosive and if you use rubber tubes, stoppers and joins in your distillation apparatus, then these will be eaten away very quickly by the hot nitric acid vapors.
You can make red fuming nitric acid from concentrated H2SO4 and KNO3 at normal air pressure, but that is not pleasant at all. The red fuming acid is horribly impure, it contains a lot of dissolved NO2 and hardly is useful for synthetic purposes.
Best thing to do is to use dilute H2SO4 (e.g. 60%) and add KNO3 to that. That will give water at first, but at a certain point the azeotropic mix of water and HNO3 (containing 68% acid by weight) will distill. In this way, you'll get an almost colorless acid. The first part is mainly water, the last part is the azeotropic mix. When all of the distilled liquid is taken, then with 60% H2SO4 I expect acid of 40 - 50%, but you need to perform precise math to get the precise concentration. This acid is perfectly suitable for dissolving lead metal. For dissolving lead metal you need 40 - 50% acid, higher concentrations may lead to passivation and very slow dissolving of the metal.
If you have 33% H2SO4, then boil it somewhat, such that you get 60% acid. You do not want to boil it all the way down, such that you get 96% acid.
You cannot use HCl. When that is mixed with nitrate, then you get an aqua regia like liquid. The following reaction occurs on heating: KNO3 + 4HCl --> NOCl + Cl2 + 2H2O + KCl. The NOCl and Cl2 escape as (very toxic and corrosive) gases. You can recognize the formation of the NOCl and Cl2 by change of color. The liquid becomes yellow/orange and a yellow/orange gas mixture is formed.
A word of warning with all these experiments. Distilling HNO3 is a very risky thing. It can be done at home, but if you have never done a distillation before, then first try distilling something less risky, e.g. some water/salt mix, just to get the idea of how this works.
When you dissolve lead metal in HNO3, then initially the reaction seems not to run at all. A little heating may be needed. But, once the reaction runs, a lot of brown NO2 gas is formed and a crystalline white mess is formed in the acid. The NO2 gas is insidiously toxic and has delayed effects. At the moment of breathing you hardly notice its effects, but it may cause lung edemia after 15 hours or so. Be very careful and do this outside. Another serious risk is that when the lead is dissolved, many small droplets of the liquid are sprayed into the air. These droplets consist of HNO3, water and Pb(NO3)2. The HNO3 and water quickly evaporate and the Pb(NO3)2 remains as tiny dust specks. This also is very harmful for you health. So, if you perform the reaction, cover the beaker loosely with a paper tissue. This will capture small droplets of liquid, while the gas (NO2) will escape. Lead-salts are a cumulative poison. One exposure is "remembered" by your body and that is the reason why you need to be so careful with the lead salts. All this experimentation MUST be done outside on a windy day, or in a good fume hood.