How do I make a mob riding another mob and make the rider invisible?

When you're debugging commands, always make sure to edit them in an external editor, so you can have proper indentation. This way you can immediately see the errors.
This is your command:

/summon blaze ~ ~5 ~ {
    ActiveEffects:[
        {
            Id:14,
            Amplifier:1,
            Duration:999999,
            {
                Riding:{
                    id:"Bat"
                }
            }
        }
    ]
}

As you can see the Riding tag is still part of the compound list for active effects, which makes no sense. If you move that to the correct location:

/summon Blaze ~ ~5 ~ {
    ActiveEffects: [
        {
            Id:14,
            Amplifier:1,
            Duration:999999
        }
    ],
    Riding:{
        id:Bat
    }
}

The command works perfectly fine. Here again in one line:

/summon Blaze ~ ~5 ~ {ActiveEffects: [{Id:14,Amplifier:1,Duration:999999}],Riding:{id:Bat}}