Why do you think nominal housing prices will not increase with the new money? I can understand why housing prices would be sticky when going down, but I've never heard that they are sticky when going up.

Increasing the housing supply would tend to cause nominal housing prices to decrease. Increasing the money supply would tend to cause nominal housing prices to increase. Do them both at the same time and they cancel.

Ah, reading comprehension failure on my part. Wouldn't the effects of the inflation be out of proportion with the housing supply increase, though? Printing money will affect many prices, but the housing supply only affects the local housing market.

Sure, the nominal prices of everything else would increase, as would wages so that purchasing power would remain largely unchanged. Moderate inflation is basically harmless -- in a lot of ways it's beneficial. It's effectively a tax on money (especially on lenders), which makes it extremely progressive. (That is one of the reasons deflation is so catastrophic. The other is that deflation makes cash an investment, so people would hoard cash instead of investing in actually productive activities.)

And the government can use the new money in place of normal tax revenue, which helps the economy.

It's possible to go too far and have hyperinflation. It becomes problematic if prices double every day and nobody will accept your money because it will have lost half its value by the time they get to the bank, as is often the case in failing or mismanaged countries. But if we had e.g. ten percent inflation per year for several years, the effects would be a rather large net positive. It's basically a wealth transfer from lenders to borrowers, where "borrowers" means the taxpayer (national debt), people with student loans or car loans or credit card debt, etc. And homeowners with mortgages, canceling the reduction in housing prices that would otherwise come from significantly increasing the housing stock.