If you have several measurements of the same quantity several times in the two units, there is, in general, no way to estimate the transformation from one unit to the other.
However, if you knew that that there is a multiplicative relationship between the two, and that the noise in the two sets if measurements is zero-mean normal (with equal variances or different but known variances), then you can estimate the multiplicative factor $k$ by maximum-likelihood.
If you make the above assumptions you can proceed as follows. Let $X_B$ be the actual value of the quantity you repeatedly measure in units of $B$. Then $L_{Ai} = k X_B + e_i$, $i = 1, \dots, n$, and $L_{Bj} = X_B + f_j$, $j = 1, \dots, m$.
$e_i$ and $f_j$ are normal i.i.d., normal random variables with mean 0 and variance $\sigma^2$. You can write the log-likelihood of the data as
$$ L(data; k, X_B) = const - \frac{1}{\sigma^2}\sum_i (L_{Ai} - k X_B)^2 - \frac{1}{\sigma^2}\sum_i (L_{Bi} - X_B)^2 $$
You should be able to maximize this quantity in terms of $k$ and $X_B$ to obtain your transformation (and an estimate of the quantity).
In fact, if you go through the algebra of setting the partial derivatives of the log-likelihood function with respect to $k$ and $X_B$ to zero, you should get the expression for $k$ you have in your question.
$X_B = \frac{\sum_j L_{Bj}}{m}$ and $k = \frac{ m \sum_i L_{Ai}}{n \sum_j L_{Bj}}$