At some point it ran out of securities that its (commercial) counterparties would accept as collateral for these very short term cash loans. And that's where apparently the Fed took over, by taking in junk in exchange for fresh cash.

It appears that's not correct, and may be an exaggeration by the New York Times when they say

They were considered the dregs of Lehman Brothers -- "bottom of the barrel," as one banker put it. But as Lehman executives tried to keep the floundering bank afloat in 2008, they used these troubled investments to raise quick cash that helped mask the extent of the firm's troubles. And they did it with the help of the Federal Reserve Bank of New York.

If Lehman brothers an out of willing counterparties it was not because of the quality of the assets, but because its own creditworthiness was deteriorating, and because of the high value of liquidity. Specifically, according to FT Alphaville's What's in Repo 105

But what exactly was Lehman Bros stuffing into the Repo 105 sausage?

Perhaps counter-intuitively it was not using the stuff on its balance sheet that was hardest to sell into markets.

Rather, it was the most liquid -- things like A- to AAA-rated securities, Treasuries and Agency debt, which you can see in the below table, from the Examiner's Report (Appendix 17)

What this means is that Lehman was using Repo 105 exclusively to reduce its leverage (by shrinking its balance sheet) rather than to get liquidity. It needed to put good quality collateral into the Repo 105 in order to find willing counterparties. Of course, as the crisis worsened it had to turn to the Fed, but the collateral seems to have been good (or, at least, blessed with a good rating by the agencies) except for a 5-10 percent of the total value of the Repo 105. Tyler Durden of Zero Hedge writes in The "Repo 105" Scam: How Lehman Fooled Everyone (Including Allegedly Dick Fuld) And How Other Banks Are Likely Doing This Right Now

In August 2008, just before it was over, the firm allowed $55 million, or seven securities, rated CCC to be included in a Repo 105 transaction.

$55 million is just 0.1% of the $50bn that Lehman shifted off of its balance sheet in the second quarter of 2008 alone (link to HuffPo). Tyler Durden proceeds:

The next chart makes it evident it that 105s were used simply to game the firm's assets into quarter end (yellow highlights), by reducing overall asset for leverage ratio calculations.

That this scam was going unsupervised (just who the hell were the counterparties?) for many years, and that many banks are likely using it right now to fool investors, regulators, rating agencies, and the idiots at the FRBNY (who certainly also know about this), is beyond criminal. Yet that nobody will go to jail for this is as certain as the market going up another 10% tomorrow. A full investigation has to be conducted immediately into whether existing Wall Street firms, and in particular those who use Ernst & Young as auditors, are currently abusing public confidence via such transactions.

The "scam" is not that they were getting cash for junk, but that they were reporting repos as sales, not as financial transactions.

The brainless should not be in banking -- Willem Buiter