Lightning Network (LN) was predominantly devised as a way to scale Bitcoin (in terms of both throughput and keeping tx costs low). Some people also throw in "privacy", but that has been largely debunked, e.g. 1.
Monero doesn't have these problems, hence there is not some great need for LN with Monero.
Another much trumpeted "feature" of LN is the potential for cross-chain atomic swaps. However, this has also come up against some criticism, for example 2.
Leaving aside the questionable needs / benefits of LN, the question isn't really what or not can be "ported" due to the fact Monero is not a Bitcoin clone, but rather: is it possible to implement LN for Monero. And the answer to this right now is no. LN requires time-lock and refund mechanisms to operate. Whilst some kind of time-lock trickery could perhaps be achieved, due to the way the blockchain and transactions operate, there is no refund mechanism currently.
... what parts of Lightning would need to be rewritten for Monero?
Monero would need non-trivial changes to be able to use the LN scheme. Then any LN daemon would need work to implement on top of the Monero blockchain. Thus "what parts" is actually a lot of both.
Would the Lightning Network Daemon be compatible with minimal changes, for instance?
What are some things that wouldn't be compatible?
Given how tightly any 2nd layer needs to operate with any base layer, there's a considerable amount of incompatibility.
Monero has been unable to directly port much of the tools and code written for Bitcoin and its forks
You say this like it's some kind of negative. There are many issues with Bitcoin's (and its forks) design and code. Monero is not a fork of Bitcoin for good reason.