For the love of everything DO NOT inject yourself with air, air bubbles in your veins are EXCRUCIATINGLY painful; it CAN be lethal, but its unlikely unless you know what your doing.
However for washing detergent there is lots of stuff in it, and if you want to how lethal it is, you should look at all the ingredients in it, and look at how they would interact with eachother. You should also know how each chemical will dissolve in your blood, since with SN conventional, pure chemistry says (from what i learned) It should just dissolve, but it causes hypoxia when you look at it biologically. So look at each detergent you have access to and research all the chemicals they have in them and see how they all will affect your blood, and your system as a whole.