Before I begin with how to fix this... We do not have 'radically low taxes'. We are about 26% of GDP. The UK is 34%. The France, Germany and the Nordic countries are insanely high in the mid 40's. Switzerland is about 28%
That said... here is what I would propose:
1) Eliminate the tax code as it stands... all of it. All 75k pages of it. All loopholes, all deductions, all subsidies. For the individual side: Replace it with a Flat tax of 20% on earned income from 0-$500k Additional 10% tax on anything over that. This is on ALL income: Earned income, dividends AND capital gains. Standard deduction for every adult is $30k. Which will make the tax code progressive and protective of the lower income families. For the corporate side: recognize that a corporation is just paper. Eliminate the corporate tax code all together. The individual tax code will take care of taxing the money based on who gets it.
2) Spending: We must get this under control, obviously. You pinpoint the obvious first place to start... The DoD. Way too much waste there. Let Europe pay to defend Europe. Eliminate most of the bases there, eliminate the bulk of the top of military hierarchy (we have far too many Generals/Admirals etc... who all have staff... basically too much admin). Definitely stop spending on weapons programs the military doesn't even want/need. Bottom line, I agree we could cut Defense by at least $200B.
3) Healthcare costs: It doesn't matter how we pay for healthcare if we don't address the cost issue. What is driving the costs so high? Why is it so expensive to get a medical degree? Why are insurance costs so high for doctors/hospitals? I have more questions than answers here, but this is first place to start.
4) Social security: Fix it with an additional 3% tax on income over $500k. That would resolve the issue. Social Security is not nearly as big of a problem as people think. Medicare is... which is due to number three above.
All in all, I think we would agree on a lot... Not much we can do with the amount we have to pay out of SS, but we need to balance it with an increase in taxation as mentioned so that it doesn't detract from other areas in the future. Besides SS, Healthcare via medicare and medicaid is the biggest mandatory expense. We must figure out a way to get costs under control. Defense is the non-discretionary target. I think we agree that at least $200B, maybe more, could be cut.
I think the area for discussion for the two of us would be on taxation. I want more transparency and simplicity in the tax code. There shouldn't be 75k pages to go through. My plan above is certainly open for tweaking on the numbers... but I think it is the better option than what we have today.
Increasing taxes will fix NOTHING.

Willful idiot thinks billions will solve deficits in the trillions.