Aligning Data frame with missing values

I'm using a data frame with many NA values. While I'm able to create a linear model, I am subsequently unable to line the fitted values of the model up with the original data due to the missing values and lack of indicator column.

Here's a reproducible example:

library(MASS)
dat <- Aids2
# Add NA's 
dat[floor(runif(100, min = 1, max = nrow(dat))),3] <- NA
# Create a model
model <- lm(death ~ diag + age, data = dat)
# Different Values
length(fitted.values(model))
# 2745
nrow(dat)
# 2843