About 0.9% of the US federal spending goes to treatment of End Stage Kidney Disease. The most recent data available when the episode aired give a value of 0.94%, which could reasonably be described as "nearly 1% of the federal budget".
Of note, this is not all federal spending on kidney disease just spending on treatment for end stage disease, which is defined as chronic irreversible kidney failure that requires either dialysis or transplant to treat. There is additional medicare spending for chronic kidney disease treatment and kidney disease research, but that's not involved in Last Week Tonight's claim.
The ESRD episode of Last Week Tonight aired in May of 2017, citing an article published online in July of 2016. As mentioned in the comments on the question, the article references a broken link. Lets look at the available data from the Congressional Budget Office (CBO) and US Renal Data System (USRDS) numbers in 2015 and 2016 to evaluate the claim in the context it was made.
Each year, the USRDS publishes a report that includes total federal spending. These reports include data from up to two years prior. So in 2015, 2013 numbers were available. In 2016, the most recent numbers were from 2014. The USRDS page that links to these excel files are here for 2015 and here for 2016.
In 2013, total federal spending on End Stage Renal Disease (ESRD) was 31.8 billion. In 2014 it was 32.8 billion (see linked excel documents above).
Total federal outlays in 2013 were 3.455 trillion. 3.506 trillion in 2014.
In 2015, then, the most current data showed federal spending on ESRD to be 0.92% of total federal spending. In 2016, the most current data showed federal sending on ESRD to be 0.94% of total federal spending.
For reference, the most current data is from 2017, which shows federal spending on ESRD to be 35.9 billion. Total 2017 federal spending was 3.981 trillion, making ESRD spending 0.90% of total spending.
Checking my work
The most convenient way to get these numbers are to look at this table from the USRDS and this document from the CBO. Both tables include all the years discussed above. In my discussion above I linked to the specific reports available at the time the claim was made in order to give proper context, but all the data is available in these two documents.