This is one of those builds that is crying out for 3E multi-classing spellcaster advancement. Assuming you take Eldritch Knight for the Fighter archetype, those 6 levels in Fighter should give 2 levels of spellcaster advancement, stacking with Wizard 5 for a total that should be equivalent to Wizard 7.
Likewise in 3E, the Wizard level 5 has +2 base attack bonus, which would stack with +6 form Fighter level 6, for a total of +8. This is moot in the 5E context, as proficiency bonus scaling doesn't work like that. But maybe it is relevant if your DM were to permit the 2nd extra attack at a total equivalent to Fighter level 11. So, if you were say a Fighter 7/Wizard 8, if that stacks as Fighter 7 + 8/2 = 11 total equivalent in Fighter advancement, that's where you'd get the 2nd extra attack.
This is essentially along the lines of what Horwath is suggesting I gather.
More broadly, many evenly balanced multi-classed 5E builds are in need of a better mechanism of keeping them at a reasonable progression in their core combat skills. There are a few that can synergize well, if you build them carefully, e.g., I had a Barbarian 8/Rogue 7 that worked quite well with Strength 20, dual-wielding finesse weapons.