How do I calculate the random effects of the model? I need to work out how to calculate the random effects from my random intercepts multilevel model, to test if wellbeing of workers (level-one) is affected by the hospital they work in (level two). 
My fitted equation looks something like this:
WBSCOREij= β0CONST+ β1HOURSij + Uj + Eij
For an individual who works 15 hours in hospital 5, this means:
WBSCOREi5 = 1.886 + 0.073*15 + U5 + Ei5
WBSCOREi5 = 1.886 + 1.095 + U5 + Ei5
How do I calculate the random effects of the model?  I don't know the wellbeing scores of all the individuals from hospital 5.
But I do know that MLwiN gave me estimated outputs for Uj as 0.037 and for Eij as 0.347.
Do I already have my answer?
Is the wellbeing score of a worker just simply:
WBSCOREi5 = 1.886 + 1.095 + 0.037 + 0.347
I was under the impression that the estimated Uj and Eij were for ALL cases, and not just for hospital 5.
Any advice would be most welcome, thank you!
 A: $u_j$ is the deviation from the average predicted score due to the influence of which hospital someone was in. Just like $e_{ij}$ is the deviation from the averaged predicted score of someone in a specific hospital for an individual. $\sigma^2_{u_j}$ is the variance of the intercept. 
Each hospital has its own intercept. Your output (I'm nearly certain, but it would help if you posted the output from the model you ran) means that the average intercept is 1.89, with a variance of .037. That variance shows that it is a random effect, and thus you have already "calculated the random effects."
As far as predicted scores go, you can slice it a couple of ways. I'll stick with keeping hours worked constant, saying we want to predict well-being for someone who worked 15 hours.
The well-being of any particular person in the sample—who worked 15 hours—would best be predicted by (that is, the expected value of $y_{ij}$, the well-being score):
$WBSCORE_{ij}$ = 1.886 + 1.095 + $u_j$ + $e_{ij}$
Now, each hospital has its own intercept. So you can add on another constant, which varies by hospital. If you are using the lme4 package and have fit an lmer() object, you can extract these random effects for a model (let's give it the name mod1 for now, and the cluster variable called hospital) by entering:
coef(mod1)$hospital

Let's say you want the predicted score for the average person in Hospital 12. You can check that output and, let's say you find that Hospital 12 has a random intercept value of -0.20. You would simply fill that in the place of $u_j$, as in this case $u_j$ = $u_5$ = -0.20. So you have the equation:
$WBSCORE_{i5}$ = 1.886 + 1.095 - 0.20 + $e_{ij}$
So, to address, your last point, no, the expected value of well-being score is not:
$WBSCORE_{ij}$ = 1.886 + 1.095 + 0.037 + 0.347
Those last two terms are variance estimates (again, to be 100% certain, I would have to see the output) and are random (or residual) terms whose expected value is 0:
$E(y_{ij})$ = 1.886 + 0.073*$HOURS_{ij}$ + $u_j$ + $e_{ij}$
Since the expected values of the last two are 0, you simply have:
$E(y_{ij})$ = 1.886 + 0.073*$HOURS_{ij}$
