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According to this blog post from 2013, the MuMIn package in R can provide R$^2$ values for mixed models ala an approach developed by Nakagawa & Schielzeth 2013$^1$ (which was mentioned in a previous answer).

#load packages
library(lme4)
library(MuMIn)
 
#Fit Model
m <- lmer(mpg ~ gear + disp + (1|cyl), data = mtcars)
 
#Determine R2:
r.squaredGLMM(m) 

       R2m       R2c 
 0.5476160 0.7150239  

The output for functionr.squaredGLMM provides:

• R2m: marginal R squared value associated with fixed effects

• R2c conditional R2 value associated with fixed effects plus the random effects.

Note: a comment on the linked blog post suggests that an alternative Nakagawa & Schielzeth inspired approach developed by Jon Lefcheck (using the sem.model.fits function in the piecewiseSEM package) produced identical results. [So you have options :p].

• I did not test this latter function, but I did test the r.squaredGLMM() function in the MuMIn package and so can attest that it is still functional today (2018).

• As for the validity of this approach, I leave reading Nakagawa & Schielzeth (2013) (and follow-up article Johnson 2014$^2$) up to you.

^{1: Nakagawa, S., and Schielzeth, H. 2013. A general and simple method for obtaining R2 from generalized linear mixed‐effects models. Methods in Ecology and Evolution 4(2): 133-142.}

^{2: Johnson, P. C. D. 2014 Extension of Nakagawa & Schielzeth’s R2GLMM to random slopes models. Methods in Ecology and Evolution 5: 44–946.}