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 and estimate of the quantity).