Since you mention in a comment that you would be the first to work remotely in your organization this is going to be extra tough I think. It's not impossible by any means but I think you need to be prepared to respond to likely objections:
Won't the kids be a distraction?
Claiming that they simply never will be is likely to come across as incredibly naive. Most kids who aren't old enough to be left unsupervised aren't up to going all day, every week without requiring some of your attention. Little Johnny won't share with Little Billy or Little Billy hits Little Johnny etc etc.
If you are somehow able to convince your employer that they will never distract you when working from home then I'd expect the to wonder why you need to be there at all.
The best approach IMO is to acknowledge the realities of the situation and provide a resolution. Say that while your kids are old enough to understand that you are there to work not spend time with them you know that there will inevitably be the occasional disruption and that you're prepared to work extra time when that happens to make up for the lost time.
Why can't your wife do it?
Sadly this will be an all-too common objection, either through old fashioned gender-role attitudes or a sense of a "prior claim" in that they had you working Wednesdays first. Explain that the old system/schedule was no longer sustainable for your family and that while it wasn't the hoped for outcome your wife was unable to secure a working pattern that allowed her to remain at home on the Wednesdays.
How will we know you're actually doing any work?
For companies that have never done WFH before the fear that employees will just be treating it as a psuedo-day off is very common. So have some solutions ready to offer them for how they can track your progress and "presence". Whether that's source control commits, or IM or e-mail reports etc, give them some options but remind them that you're happy to work with whatever suits them - the whole point is to make them feel comfortable.
What if we need you for a really important meeting on a Wednesday that can't be rescheduled?
If you can, try and have some back up plans in place - even if they are things that you can only call on occasionally (neighbor, childminder, family, wife uses a day off, etc).
Overall I think the better prepared you can be to demonstrate how you are committed to ensuring this creates the absolute minimal level of disruption to them the more likely you are to get the "yes".
