A query rather than a straight answer. It is something of an unfortunate historical accident that early formal systems of logic (Frege, Russell/Whitehead, Hilbert) went for many-axioms/few-rules. These systems are pretty unnatural, and indeed in the background there is arguably a mistake about what logic is about. For the early founding fathers tended to speak as if logic was about cataloguing logical truths, rather than valid proofs -- take the latter view, and you'll conceive of the natural way of presenting a logic as a system of rules of inference which you can use in constructing mathematical and other proofs, rather than as a system of logically true axioms from which more logically true propositions can be derived.
So I do wonder why, other than for a somewhat pointless technical exercise, we should nowadays be interested in presenting a many-axioms/one-rule system of logic for negation/conjunction, rather than a much more natural no-axioms/many-rules system? And for the latter, lots of standard textbooks deliver the goods for free. To get a complete natural deduction system for negation and conjunction just take the system for the usual four or five connectives and leave out the rules for disjunction and the (bi)conditional.