You were interested to see the links below.

https://community.jmp.com/t5/Discussions/How-to-measure-the-effect-size-of-fixed-effects-in-a-mixed-model/m-p/11315#M10852

https://docs.google.com/document/d/1TXbqYGfeHTuk69Sl7btn9EC6jsMrAo9Ss-qtIDPJNE4/edit

There is no consensus on how to calculate standardised effect sizes for linear mixed models. More generally there are lots of models for which there is no such consensus (e.g., generalised linear models). In a standard linear model standardised effect sizes are standardised using the variance or SD of outcome (d family) or outcome and one or more predictors (r family). These are constant in a standard linear model. In a linear mixed model (and some others) the variance is not constant (it depends on the value of the predictors and what is in the model). One can create analogs of things like R^2 but they don't necessarily have the same properties of R^2 in a linear model or the same interpretation.

My preference is to interpret the b coefficients directly (as unstandardised effect size) and that's arguably a better approach even in a simple linear model. You can get d family analogs by using the SD of the outcome in the raw data. For example of the estimated difference is 1.8 and the SD in the raw data is 3.6 your d is 0.5 (half a standard deviation). There are lots of proposed R^2 measures. A relative simple one to interpret is the correlation between model predictions and observed data this gives a multiple R type measure and its square is an R^2 type measure. (I like it because its easy to interpret but is is also easy to show that it might depends on the range of there data and tends to be low for discrete outcomes etc.). There are no perfect measures of this type (in my view).