Ok, here is my curve: $y^2 = x^3 + 15$
I have a point, $P$ at $(-2.21, -2.051)$ and I want to get to $2 * P$,
There is an online calculator: (https://andrea.corbellini.name/ecc/interactive/reals-mul.html) and the calculator says $2 * P$ is $(17.178, 71.304)$
I can get the slope with: $\frac{3x^2 + A}{2y}$
and I can get $X$ with: $s^2 - 2x$
However, I'm unable to find $Y$, what should I do?
Your curve has distinct roots, so we can proceed in the most straighforward way. There is a subtlety that is important but easy to forget: The addition law for elliptic curves is given by $P_1+P_2 = P_3=(x_3,-y_3)$, where $(x_3,y_3)$ is the point on the curve where the line through $P_1$ and $P_2$ intersects the curve at a third point.
If $P = (x_1,y_1)$, $2P = P+P = (x_2,-y_2)$ then the line tangent at $P$ can be found to have slope $m=f'(x_1)/2y_1$ (where $y^2 = f(x) = x^3+15$) by implicit differentiation. The line is then $y-y_1=m(x-x_1)$ so plugging in the $x$ value you obtained, say $x_2$, gives you a third point of intersection \begin{align} y_2 = \frac{f'(x_1)}{2y_1}(x_2-x_1)+y_1. \end{align} Then $2P = (x_2,-y_2).$
You can check out Koblitz's book cited below for more information, particularly section I.7 The Addition Law.
Koblitz, Neal, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, 97. New York etc.: Springer-Verlag. viii, 248 pp. DM 112.00 (1993). ZBL0553.10019.