What were the types of currency used in ancient Mesopotamia?
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First, it is important to understand that the economic system of ancient Mesopotamia was something much closer to a barter system than a modern market. Money did exist, but not in the fully standardized form we are used to today. Here is a relevant article which explains:
Although Babylon had flourishing trading activity, Hammurabi did not come up with token money in the form of coins with a royal stamp as emerged in Lydia, Asia Minor and Greece around 650 B.C. (Mundell, 2002). However, money was in use in the form of commodity money. For example, a shekel is expressed in terms of weight. Fines and compensations in The Code are expressed in terms of weight in silver.
While I only see shekels mentioned in Harper's translation of the Code, two of the other units you mentioned are also apparently common ancient units of weight in silver. Specifically:
1 shekel = 20 gerah
1 mina = 60 shekel = 1,200 gerah
I'm not sure if it is known what exactly a shekel of silver was under Hammurabi. According to Britannica:
In one surviving form, from the Babylonian period, the mina weighs about 640 grams (about 23 ounces), while in another it weighs 978 grams (about 34 ounces).
So if my math isn't wrong, a shekel might have been something like 10.7 - 16.3 grams of silver.
The two other units you mention (gur and ka) are a bit different, at least within the Code of Hammurabi, as they are only used as units of grain, or in one instance, drink. Here is an amateur attempt to work out general equivalencies between grain and silver in the Code, but as you mention, it all gets pretty confusing.
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Edit in response to comment from @LangLangC.
The role of barter in ancient economies in general and in Mesopotamia has been prominently downplayed by Modern Monetary Theory. But as far as I know, these theoretical debates around the origins of money don't dispute the fundamental historical point made above: the shekel in the Code of Hammurabi is a unit of weight in silver, and this is as close as you are going to get to a single abstract unit of account in ancient Mesopotamia.
That said, the Hudson article linked to by @LangLangC does highlight a point which is extremely relevant:
A bimonetary system was created for paying the palace and temples and for valuing disparate commodities and functions, by setting the shekel-weight of silver (8 grams) as equal to a gur “quart” of grain or 300 sila. Acceptability of grain and silver for settling official debt balances catalyzed their usage as money throughout the economy.
This policy that 1 shekel of silver should be equal to 1 gur of grain apparently predates the Code of Hamurabi by about 200 years.
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