This is an easy problem to solve with analytic geometry, but not so easy with straightedge and compass alone.
We're given two perpendicular lines which meet at the center $O$ of the ellipse $\mathcal E$. The vertices of $\mathcal E$ are somewhere on these lines.
We're also given points $A$ and $B$ (not on the given lines) which belong to $\mathcal E$, how can we construct the vertices/foci of $\mathcal E$ with just this data?
I'm gonna rewrite the solution from Inteligenti pauca's link, which is more general than the problem I proposed as it considers the axes any two random conjugated diameters.
draw lines $AN$ and $BM$, both parallel to diameter 1, with $N$ and $M$ belonging to diameter 2. Let $B_1'$ be the reflection of $B$ w.r.t $M$ and let $A_1'$ be the reflection of $A$ w.r.t. $N_1$ (the intersection of parallel to diameter 2 through $A$ with diameter 1, not drawn above).
Now draw a line $MX \parallel AB$ through $M$, with $X \in AA_1'$ and draw $MY \parallel A_1'B_1'$ through $M$ with $Y \in AA_1'$.
Now take $C = AA_1' \cap BM$ and draw a perpendicular to diameter 2 through $N$ and let point $D$ lie on it such that $ND = \sqrt{CX \cdot CY}$. The circle $\odot (O,OD)$ meets diameter 2 in vertices $V_1$ and $V_2$ of the desired ellipse. To find the other vertices in my problem it is enough to do an dilation on the circle we found, but the rest of the solution works for any pair of conjugated diameters.