I am interpreting OP's question as asking why Germany did not ask Ireland to invade the UK rather than why Germany did not invade Ireland. In order to answer this question it's important to understand why Ireland remained neutral during the war.

Some of the most commonly given reasons are:

• Strong Anti-British sentiment by the Irish public and the relative recency of the Irish Civil War. Adopting a pro-British stance would have re-opened these wounds.
• By opting to remain neutral, Ireland exercised its rights as an independent nation, which was quite important for the government and its citizens due to its status as a newly formed nation.
• Ireland did not declare war on the UK because its geographic location, resources, manpower, and its defenses were completely inferior to the UK's.

Here is a good comment regarding the same question as to Ireland's neutrality during the war: https://www.reddit.com/r/AskHistorians/comments/3d0hw7/did_the_nazis_ever_seriously_consider_invading/ct30jzt/ (u/petros08)

Furthermore, Dwyer's "Behind the Green Curtain: Ireland’s Phoney Neutrality During World War II" is also an interesting read. He also holds the popular view that Irish neutrality during the war strongly leaned towards the allies and that, as in the comment above, de Valera made a couple of short-sighted decisions in order to maintain neutrality that hurt Ireland's public image in the post-war years. It is also interesting to note that there is some debate as to whether popular opinion of Irish neutrality on the allied side in WWII is revisionist and an example of post-modern diplomacy.

It's pretty hard to answer questions in the form of "why did figure X choose not to do Y?" We can only answers those definitively if we have some record of person X considering option Y, and then rejecting it.

Perhaps a better question would be: Would Ireland have made a good ally for Nazi Germany?

And that answer is: no. Here's three big reasons why:

1. Military capacity

In 1936, the population of Ireland was around 3 million people---less than 5% of the population of Germany at the time. I haven't been able to find precise figures, but the armed forces were maybe around 50,000 strong. The whole army had less than a dozen armored cars and only two tanks. The Coastal and Marine Service (CMS, the Irish Navy) had literally one ship: the Muirchú. This was not a fighting force that could provide much benefit to Hitler.

2. Popular opinion

I don't believe any opinion polling was done in Ireland until well after this time, but considering that some 5,000 Irish soldiers decided to desert and support the allies, it would seem that Germany would have a tough time convincing the Irish people to join their side.

3. No Comparison

Most of the other minor Axis powers did have good reasons to join the Axis forces. Maybe they were still trying to regain territory lost in previous conflicts (Bulgaria and Thailand). Maybe they had a flair for fascism and nationalism (Romania and Hungary). Ireland was still struggling with the embittered sides leftover from their own civil war. So there's just not a good reason for them to get involved in an international conflict.

In Summary

[This is kind of a "what if" question] (https://www.reddit.com/r/AskHistorians/wiki/rules#wiki_no_.22what_if.22_questions) which AskHistorians doesn't really like. The broad answer to "why didn't they" is usually either a) because it wouldn't have made sense or b) because they didn't know.

And it's usually a, which is kind of boring. Sorry.

---

Sources:

• Irish Army Vehicles, transport and armour since 1922. Karl Martin.
• Ireland 1868–1966. M.E. Collins.