There is a systematic graphical notation that allows the tracking of signs, which I will mention at the end of this answer. But before doing so, let me outline the situation when $2 = 0$.

Note first that since $(A,\nu)$ is strictly associative, $\mu_0 = 0$ and $\mu_1 = d_H = d : H \to H$. By convention, if I have multilinear maps $f: H^{\otimes k}\to H$ and $g: H^{\otimes l}\to H$, then I will write $f\circ g : H^{\otimes(k+l -1)} \to H$ for:
$$ (f\circ g) (x_1\otimes \cdots \otimes x_{k+l-1}) = \sum_{i=1}^k f\bigl(x_1\otimes \dots \otimes x_{i-1} \otimes g(x_i\otimes \dots \otimes x_{i+l-1}) \otimes x_{i+l} \otimes \cdots \otimes x_{k+l-1}\bigr) $$
This has a useful graphical notation, wherein the composition is the sum over all rooted planar trees with an $f$ at the bottom node and precisely one $g$ at one of the upper nodes.

Then axiom to be an $A_\infty$-algebra in characteristic $2$ is
$ 0 = \sum_{j=0}^{n+1} \mu_j\circ \mu_{n+1-j} $, or, since $\mu_0 = 0$ and $\mu_1 = d$:
$$ [d,\mu_n] = \sum_{j=2}^{n-1} \mu_j \circ \mu_{n+1-j} $$
The right-hand side is a sum over all rooted planar trees with $n$ leaves and precisely two nodes, each of which has at least two branches from it.

To check this, the first thing to convince yourself is that the operator $[d,-] : f \mapsto d\circ f + f\circ d$ is a derivation of composition and tensor, so that to apply $[d,-]$ to some large diagram, you sum all diagrams you get by replacing one component of your original diagram by $[d,-]$ of it. Note also that $[d,-]$ comutes with (i.e. annihilates) $i$, $p$, and $\nu$. So when you work out $[d,\mu_n]$, you get a sum over diagrams that look like $\mu_n$ (i.e. planar rooted trees with $n$ leaves, each node has two branches, and interior edges labeled by $h$), except one of the interior edges has been replaced by $[d,h] = \mathrm{id}_A + ip$.

Now, it should be completely clear that the diagrams where the $h$ is replaced by an $ip$ are precisely the diagrams appearing in $\sum_{j=2}^{n-1} \mu_j \circ \mu_{n+1-j}$. (If this is not clear, let me know, and I will try to make it clearer.)

Finally, we must dispense with the diagrams in which an $h$ is replaced by an $\mathrm{id}$. For any such diagram, consider contracting it along the offending $\mathrm{id}$ vertex, to produce a node with three branches. Except the resulting diagram with the trivalent vertex can be produced in two ways, corresponding to the two planar ways of blowing up a rooted node with three branches into two two-branch nodes. So, after sorting all of your offending diagrams into such pairs, you get a sum of diagrams that looks like a $\mu_n$-type sum, except one vertex is has three branches. What is this vertex labeled by? Why, $\nu \circ \nu$, of course, which is a sum of two terms. On the other hand, $\nu \circ \nu = 0$ by the associativity law for $(A,\nu)$.

In characteristic not equal to $2$, the exact same argument works, but you must find a good convention / notation for signs. The best notation that I know is as follows. It should, of course, already by understood that the solid "$H$" or "$A$" edges extend to "infinity" at the top and bottom of the page. You should additionally draw diagrams with some other color of edge (I usually used "dashed") that records the degrees of operators — so this "dashed" edge should carry an arrow denoting its direction. A vertex that raises homological degree by $n$ is required to receive $n$ dashed edges, and a vertex that lowers homological degree by $n$ is required to emit $n$ dashed edges. Free dashed edges are sent off to "infinity" at (say) the left-hand side of the page, and the order from top to bottom that the free dashed edges arrive is important. Just as you cannot add diagrams whose numbers of input and output "$H$" strands mismatch, you similarly must have the same sequence of dashed edges. (In categorical language, what I'm saying is that you only work with "global" elements of endomorphism spaces, which is to say actual morphisms in the category of homologically-graded abelian groups, but that you give yourself access to the *objects* of this category which are lines in degree $\pm 1$.)

Now whenever two edges cross, something happens with signs. The notation basically takes care of this, but if you ever insist on working with "homogeneous elements" (which is a bad habit — it's better to work more categorically) the convention is that as an element runs down the "wire" of a solid edge, when it passes through a dashed edge it remains unchanged if it is of even degree and changes sign if it is of odd degree. The notations that do matter are:

- A closed dashed circle can be removed for a factor of $-1$.
- A dashed crossing can be resolved for a factor of $-1$. (Since dashed edges are directed, any dashed crossing has a unique resolution.)

For example, the operator $d$ emits a dashed edge, and the operator $h$ receives a dashed edge. Thus an equation like "$\mathrm{id} - ip = dh + hd$" is nonsense: the left-hand side has no dashed edges running to infinity, whereas on the right-hand side the first summand emits an edge and then receives one, and the second summand does those in the opposite order. To sum the two terms on the right-hand side you have to at least get them into their edges-at-infinity into the same order, which you can do by adding a crossing (but remember that resolving that crossing changes a sign). To make the two sides agree, you get connect up the two dashed edges, and again you should think a moment about signs. The correct right-hand side to "$\mathrm{id} - ip = dh + hd$" is:

Yes, the sign is correct. (Incidentally, I'm lifting these images from my thesis, which works out a slightly different question, and so the colors and numbers are not for this post.)

Anyway, I'll leave it as an exercise to write out diagrams for $A_\infty$ algebras in this notation, and to get all the signs right. (Hint: there should be no "weird" signs.) Part of the reason that I'll leave it as an exercise is that there's not really a unique correct answer: you make a sign convention, and work with it.