"Elite Capture of Foreign Aid" is a relatively recent (and high-profile) attempt at quantifying the loss of foreign aid to "leakage".  Here's the abstract:

Do elites capture foreign aid? This paper documents that aid disbursements to highly aid-dependent countries coincide with sharp increases in bank deposits in offshore financial centers known for bank secrecy and private wealth management, but not in other financial centers. The estimates are not confounded by contemporaneous shocks such as civil conflicts, natural disasters, and financial crises, and are robust to instrumenting with predetermined aid commitments. The implied leakage rate is around 7.5 percent at the sample mean and tends to increase with the ratio of aid to GDP. The findings are consistent with aid capture in the most aid-dependent countries.

Leakage is higher than we'd like but not as big as some fear. An especially important finding is on how aid interacts with institutional strength and political economy in the recipient country:

While the leakage estimates reported above are averages for those countries with annual aid from the World Bank above 2% of GDP, we show that leakage rates exhibit a strong gradient in aid-dependence, both within this sample and beyond. On the one hand, lowering the threshold to 1% of GDP (sample of 46 countries), we cannot reject the null hypothesis of no leakage. This suggests that the average leakage rate across all aid-receiving countries is much smaller than the estimates obtained from the main sample, which account for less than 10% of all World Bank aid. On the other hand, raising the threshold to 3% of GDP (sample of 7 countries), we find a higher leakage rate of around 15%. This pattern is consistent with existing findings that the countries attracting the most aid are not only among the least developed but also among the worst governed (Alesina and Weder, 2002) and that very high levels of aid might foster corruption and institutional erosion (Knack, 2000; Djankov et al., 2008).