If, as you say, 15% of student loan borrowers were to default and the government were to forgive their loans, that would simply be a transfer of wealth from the government to those borrowers (on the order of about 15% of \$1.3 trillion, or about \$200 billion). What happened from there would depend first on whether the government funded that transfer by cutting other spending or by issuing debt. Inflation is a basically unrelated issue— the monetary authority (the Federal Reserve) is independent of the fiscal authority (Congress and the Treasury), and as they say, the monetary authority acts last.
Student loan payments are generally not due until some time after a student leaves school, so as a practical matter, forgiveness on defaulted loans would not generally affect whether students remained in school versus being in the workforce, unless you assumed that students started borrowing more with the intent of defaulting (a "moral hazard" scenario). I'll ignore this possibility just to keep things simple and to keep the focus on the fiscal and monetary impact.
In all scenarios, borrowers who received loan forgiveness would be better off, as their wealth would increase by the amount of the forgiven debt. They would, as noted, likely already be working or seeking work. Because they had already stopped paying their debt, we would not assume that they would experience improved cash flows. Work on structuring modifications to defaulted mortgages suggests that households are more sensitive to changes in cash flows than changes in net wealth, so we would only expect these individuals to increase their consumption modestly, but we would in fact expect this side of the ledger to result in increased consumption and economic growth.
However, if the government financed this debt forgiveness by reducing its consumption, it would probably be a net negative to GDP in the short run, as it would remove \$200 billion in government spending, which would not be made up by the slight increase in consumption by borrowers. Government consumption expenditures are currently equal to about \$3.2 trillion annually, which equaled about 18% of GDP in the most recent quarter, so... the direct impact would be a reduction of about 1% of GDP. You can make assumptions about consumption multipliers for government consumption and come up with different results, but this is the direct impact in that scenario.
If the government financed the debt forgiveness by issuing its own debt, it would increase the stock of outstanding US government debt by \$200 billion, or about %1. This would have little impact on debt markets, so there would be no concern about "crowding out" private borrowing, and because it was debt-financed, there would be no reduction of government consumption. As a result, the only real effect on GDP would be the small net positive contribution from increased private consumption.
With respect to inflation, even if the monetary authority bought all the additional debt issuance, it'd only increase the amount of money in the financial system by a negligible amount. For comparison, the Federal Reserve's aggregate increase in holdings of Treasury securities in response to the financial crisis and through its quantitative easing program was close to about \$2 trillion, or ten times the size of this hypothetical debt issuance. Inflation during that period has generally remained below the Fed's 2% target. So currently, the effect on inflation would likely be negligible.