How do miners determine if a wallet has sufficient funds I have ready about 10 posts that are similar and none of them seem to answer my question so any help is appreciated. 

My question is how to miners technically determine if a "wallets" has the "balance" to complete the transaction. Before you say this is a duplicate let me elaborate:

User A has private key A and Public key A.1 and A.2
User B has private key B and Public key B.1 and B.2

Transaction 1 A-> B.1 by User B sending B.1 to A and A signs the transaction (0.5 BTC for illustration) with that private Key.
Transaction 2 A-> B.2  by user B sending B.2 to A and A signs the transaction (0.5 BTC) with that same private key.

Now private Key B has 1 BTC. But sense No one knows that B.1 a and B.2 belong to B because those are derived dynamically from B how can you know how much A has? Further suppose you loose track of you public keys, so B.1 and B.2 have been forgotten. How are you able to figure out the bitcoins that B has access to?
The blockchain does not see wallets at all.  It only sees spent and unspent transaction outputs (txouts).  A txout is either completely spent or completely unspent - it cannot be "partially" spent and so there is no notion of balance.  (If you want to spend less than the full value of a transaction, you create a change output.)

In your example, there are now two txouts that B can spend.  The "private key" B would actually be a master private key - each "derived" key is really a keypair consisting of a private key B.1.priv and a public key B.1.pub.  So in order to spend the txout that was paid to B.1, user B must sign a transaction using the private key B.1.priv.  If B wants to spend the entire 1 BTC, she can create a transaction with two inputs and signed by both B.1.priv and B.2.priv.

It is true that nobody except B (and maybe A) knows that both txouts are spendable by the same person.  That is fine - they don't need to know.  This is considered a desirable privacy feature.

If B should lose B.1.pub, no problem - it can be regenerated from B.1.priv. This is part of how ECDSA works.   If she should lose B.1.priv, it can be regenerated from the master private key B.  If she loses the master private key, she is out of luck and cannot spend her coins.