1) You can't create extra time, so one option is to be stringent about when "playtime" is (with kids in the house you've obviously already got experience with these restrictions). I don't have to tell you that once agreed any such times should be sacrosanct (no "I'm too tired" on Wednesday night if that is a play night unless dying from manflu for example). Another option is to think carefully about whether any of the ton on your plate can be lifted, of course (at the expense of appearing to "side" with your wife on this matter!).

2) A "needy" partner/wife/sub/toyboy/etc. is a problem when you lead a busy life whatever kind of relationship you have! I have a strong feeling that the two often go together, i.e. one's partner can become more needy the busier one gets because they may be unconsciously saying "you are paying more attention to your work/social/whateverelseyouaredoing than to me!". Of course there are many other potential reasons such as health issues which may need to be addressed.

Finally, please think about your words "too busy to give her a response" and imagine how you would feel if you were in her situation, or your kids did that to you (actually, knowing kids it can happen!), or your boss at the school did it. In my management training they told me "make time for other people".