Symptoms are your friend, and the DM is responsible for narrating how this occurs.
Diseases and poisons which have debilitating effects on a character will normally have noticeable symptoms. In fact, the symptoms will usually be so noticeable that there won't be a check required to verify that the character is suffering from something even if you don't know specifically what it is.
Remember, rolls are for when the outcome is in doubt. If a character is feeling ill, that's not in doubt. That's something you have told them due to the nature of their disease/poison. For example: If my character is shaking uncontrollably, but otherwise feels "fine", I'm going to ask my paladin to try fixing me. No roll required, I suspect I'm suffering from something thanks to losing control of my limbs.
This is where the Paladin's lay on hands can come into play. Paladin uses lay on hands to remove a poison/disease, and sees some symptoms abate. For example: the patient is suffering from a high fever, has large pupils in a brightly lit room, is shaking uncontrollably, and their lips have turned purple. After the Paladin uses lay on hands, the lips return to their normal colour and the shaking abates, but the fever and pupils remain. This is now an indicator of a secondary effect persisting on the patient.
None of this would require rolls as the outcome isn't uncertain. They're simply observations made due to obvious symptoms.
In your case, if the two diseases/poisons exhibit similar symptoms, your Paladin shouldn't notice a change after using lay on hands. If the Paladin uses it a second time, that's when they should notice the change. As for whether or not the Paladin is aware that they've successfully removed a disease/poison with lay on hands, the text here:
You can cure multiple diseases and neutralize multiple poisons with a single use of Lay on Hands, expending hit points separately for each one.
... implies rather heavily that the Paladin is fully aware of how many diseases/poisons are currently affecting the target at the time of the feature use. Thematically I would be described it as follows:
"You lay your hands on Auriga, attempting to cleanse whatever is infesting her body bringing about her illness. A surge of power (5 HP) courses from you and cleanses some of the poison you feel coursing through her system. You consider her physical appearance and note she still looks ill. A second surge pulses into her (another 5HP from the same application of lay on hands) and you feel the darkness grasping at her vitality slither away in defeat. Her face takes on a relaxed look of comfort and she seems to be faring well, the disease/poison having been cleansed from her system."
Mechanically your player would be asking things like:
P: I use lay on hands, is she cured?
DM: No. The symptoms persist after the first 5HP.
P: I spend another 5 to cleanse any additional effect. Is she cured?
DM: Yes, the symptoms have abated.
P: Ok, I apply 10 HP to my usage of Lay on Hands.
DM: Narrates using the above.
Remember, in 5e the process is as follows:
DM describes the situation. Player describes what they want to do. DM narrates the results. In this situation, the character being healed had persistent symptoms after the first application of lay on hands, so the player would simply keep applying it to keep removing diseases/poisons until they either succeeded, or ran out of healing power to do so. It's up to the DM to narrate how it happens thematically.