How to find a function in Lambda Calculus?
In your attempt,

$$map=(\lambda a \lambda b (\lambda s \lambda d((b(st))d)))$$

$a$ plays the role of $l$, $b$ the role of $t$, $s$ the role of $k$, and $d$ the role of $x$.

So, $b(st)d$ should actually play the role of $l(kt)x$, but here we see some mismatch: $b$ does not represent $l$, $a$ does that. Also, we can not use $t$, but must use $b$ instead. 

Hence, the correct answer seems to be $a(sb)d$. We get (using the standard notational shortcuts)

$$map=(\lambda a b s d.\, a(sb)d)$$

Try to check whether $map\, l\, t$ does $\beta$ convert to the intended term.

Your original idea, namely using $kt$ instead of $k$, seems to be the right intuition, as far as I can see.