This was answered by Don Davis, 1975 Bol. Soc. Mat. Mex. In modern notation, the answer is
(at p=2) $H^* j = (A \oplus \Sigma^7 A)/I$, where $I$ is the ideal generated by $Sq^1 \iota_0$, $Sq^2 \iota_0$, $Sq^4\iota_0$, $Sq^8\iota_0 + Sq^1 \iota_7$, $Sq^7\iota_7$, and $(Sq^{(0,1,1)} + Sq^{(4,2)})\iota_7$, in Milnor basis notation. If you prefer the admissable basis, this last operation on $\iota_7$ is $
Sq^6 Sq^3 Sq^1 + Sq^7 Sq^2 Sq^1 + Sq^7 Sq^3 + Sq^8 Sq^2 + Sq^9 Sq^1 + Sq^{10}$.

I derive this quite explicitly and elementarily, as a sample of how to use some sage code Mike Catanzaro wrote, in a talk I gave at Northwestern, "http://math.wayne.edu/~rrb/papers/jnwuhand.pdf", where I also give more precise citation of Don Davis's paper.

Briefly,

you observe that $\psi^3 - 1$ must induce $Sq^4 : H^* \Sigma^4 ksp \to H^* ko$,

you compute the ker and coker

compute that $Ext^1 = F_2$, and that the split extension would give the wrong homotopy,

compute the non-trivial extension using a cocycle defining it.

There, I also note that the Adams spectral sequence for this is quite clean: a marvelous $d_2$ clears out all the rubbish, followed by a simple sequence of differentials which follow from the Leibniz rule, to give the groups of order the 2-adic valuation of 8j in dimensions 8j-1.

Added:
I should say, modulo the conjecture that a certain homomorphism really does induce $d_2$, I show this. I do believe it is correct.

Also, there is an interesting module over A(2) that comes up in the j/2 case. The module exhibits periodicity, $\Omega^4 M \simeq \Sigma^{12}M$, and has Ext exactly equal to the associated graded of the homotopy of j/2, in filtrations 2 and up. It is defined over $A(2)$.