As the problem is currently formulated, you have a binary decision (whether to buy the cards or not) and a single state variable (your current wealth). I'm assuming the deck is reshuffled every play.
A policy function will be a function $f: \mathbb{R} \rightarrow \{0, 1\}$ that will say whether to buy or not buy the cards as a function of your wealth. It takes your wealth (a real number) and maps it to a binary outcome of 0 or 1 denoting whether to play or not.
How to find a sensible $f$? If your preferences over risk satisfy the Von Neumann-Morgernstern axioms, those preferences can be represented by an expected utility function. You can then compare the expected utility of buying the cards vs. not buying the cards and find which is higher.
As an example, I will use $u(x)=\log(x)$ which is the same objective of the Kelly criterion.
Expected Utility Theory
Let $w$ be a scalar denoting your wealth, $X$ a random variable denoting the sum of the three cards, and $u(x)$ a Bernoulli utility function whose curvature formalizes a notion of risk aversion.
For $w=1000$, we can simply compare $u(w)$ with $\operatorname{E}[u(X+w-15)]$ where $u(x)$ is perhaps something like $u(x) = \log x$.
How much would we be willing to for this gamble?
Find the price $p$ such that you're indifferent between having wealth $w$ and playing the game. Solve for $p$ in:
$$u(w) = \mathrm{E}[ u(w + X - p) ] $$
MATLAB code to do this:
cards = ones(4,1) * (1:13); % Matrix of all the cards
deck = cards(:); % vectorize it
three_card_hands = combnk(deck, 3); % all combinations of 3 card hands
X = sum(three_card_hands, 2); % sum across columns to get a vector with all 3 card sums
w = 1000; % set initial wealth
myfun = @(p) mean(log(X+w-p)) - log(w)
p = fzero(myfun, 15)
I compute that at a wealth of \$1000 and using log utility you'd be willing to pay 20.9798 for the 3 cards. At a wealth or $w=50$, you'd still be willing to pay 20.3 for the cards. If you had only \$15, you'd still buy the cards because $\log(15) < \operatorname{E}[\log(15 + X - 15)]$. You always buy!
A useful, related concept is the certainty equivalent which I discuss in this answer.
Classic Kelly criteria
Kelly betting is to choose a bet size to maximize the expected growth rate of your wealth (i.e. your geometric average return). You can show this is equivalent to maximizing expected log wealth (i.e. equivalent to a log utility function). (Quick argument here)
(Digression: maximizing your expected wealth typically leads to solutions where you bet everything on any positive expected value bet but go broke almost certainly. That's horribly dumb! Expected wealth is a poor objective. Kelly's insight is that maximizing the expected logarithm of wealth has far better properties)
The classic Kelly betting question of what fraction of your wealth to bet is a different question than the one presented by the original poster. For fun, we can solve for the Kelly fraction $\alpha$. Our wealth is $w$ so our wager in dollars is $\alpha w$:
\begin{equation}
\begin{array}{*2{>{\displaystyle}r}}
\mbox{maximize (over $\alpha$)} & \operatorname{E}[\log(w + \alpha w (X - 15))] \\
\end{array}
\end{equation}
Solve this with MATLAB code:
a = fmincon(@(a) -sum(log(w+a*w*(X/15-1))), 1)
I find optimal $\alpha^* \approx 1.2488$, that is, the Kelly fraction is to spend $\approx 124.88\%$ of your wealth each bet.
Note this can never take you below zero, because you can lose at most $\frac{12}{15}=80\%$ of your bet when you draw three aces. If the worst case happens of drawing three aces, the leveraged bet of 124.88% (implicitly borrowing at 0% interest rate), then you lose 99.9% of your wealth! In the median case, you add 50% to your wealth. (Yes, Kelly betting is quite aggressive.)
Risk of ruin? (Monte-Carlo simulation)
In a quick Monte-Carlo sim I wrote of 1 million runs of up to 2000 plays, the minimum wealth I ever had was 960, and the minimum ending wealth (after 2000 plays) was 11533.
The game is fantastically attractive at a price of 15.
Thinking more broadly
As @Attack68's answer describes, you almost certainly want to bring in classic, trading risk control techniques.
A real risk here is counterparty risk or risk the game is somehow different than what I've modeled here. (eg. why the heck would your counterparty play this insane game? Are you getting scammed?) If you lose more than \$40 playing this game, it raises serious questions that something is wrong and you're not playing the game modeled.
It can be useful to have risk controls robust to model misspecification. Playing an aggressive strategy that assumes your model is exactly correct can be quite dangerous. Wall Street has many stories of funds/firms blowing up due to risk outside the model.
(Update: 2021; I corrected a typo in the Kelly section.)
