Learn Finnish: Minulla ei ole lainkaan unilääkkeitä. - I do not have any sleeping pills at all.

Not in meaning. Both translate to "at all" and are negative polarity adverbs, meaning they are only used in negative sentences. Lainkaan sounds slightly more formal or literary than ollenkaan, but they are completely interchangeable here. You could just as easily say Minulla ei ole ollenkaan unilääkkeitä.

Could lainkaan be placed at the very end of the sentence instead?

Yes! Saying Minulla ei ole unilääkkeitä lainkaan is perfectly correct. Placing it right after the negative verb phrase (ei ole) just puts slightly more emphasis on the "not at all" aspect earlier in the thought, which is very common in spoken Finnish.

Just as a refresher, why is unilääkkeitä in the partitive plural rather than the nominative plural?

Because this is a negative possession sentence. The rule is that in negative "I do not have" constructions (minulla ei ole), the thing you don't have must be in the partitive case. Since we are talking about pills in the plural, it takes the partitive plural ending.

What is the base form of unilääkkeitä, and why does it have a double -kk- here?