Euler Schemes in Stochastic Differential Equations

The answer seems easier than expected. Before answering let me also note that $bX_{n+1}\Delta W_n$ will not converge to Ito as the discretization grid becomes finer.

So the answer is that the discretisation should become $X_{n+1}=X_n +aX_{n+1}\Delta t +bX_n \Delta W_n$.

To follow-up on your answer, if you look at the implicit schemes in Kloden's Numerical Solution of Stochastic Differential Equations you will notice that all of the implicit schemes are "semi-implicit" where the only implicit part is on the drift (or deterministic) term. This is because of the following:

We also saw in Section 8 of Chapter 9 that difficulties can arise in applying fully implicit schemes to obtain strong approximations of solutions of stochastic differential equations, because they usually involve reciprocals of Gaussian random variables which do not have finite absolute moments. Consequently, finite absolute moments generally will not exist for fully implicit strong approximations and a strong convergence analysis would not make sense. For mainly this reason we shall restrict our attention her to "semi-implicit" strong approximations, which we shall call implicit

So this is actually a more general idea that the proper implicit form of a method for SDEs is for it to be semi-implicit by only making the drift term implicit. The diffusion term should be kept the same to make sure there are finite moments.